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

Innermost 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 one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Polynomial Ordering

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳Dependency Graph

Dependency Pairs:

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

Strategy:

innermost

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 10

                 ↳Polynomial Ordering

Dependency Pair:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 11

                 ↳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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

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