Term Rewriting System R:
[X, Z, N, Y]
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.

`   R`
`     ↳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(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nfrom(X)) -> ACTIVATE(X)
two new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ns(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(ns(X)) -> ACTIVATE(X)
three new Dependency Pairs are created:

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
two new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
two new Dependency Pairs are created:

ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
three new Dependency Pairs are created:

ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳Polynomial Ordering`

Dependency Pairs:

ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))

There are no usable rules for innermost w.r.t. to the implicit AFS 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)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 9`
`                 ↳Dependency Graph`

Dependency Pairs:

ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 10`
`                 ↳Polynomial Ordering`

Dependency Pair:

ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(n__from(x1)) =  1 + x1 POL(ACTIVATE(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 11`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(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) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes