Termination Proof using AProVE

Term Rewriting System R:
[X, Z, N, Y, X1, X2]
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FROM(X) -> CONS(X, n_{_from}(n_{_s}(X)))
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> ACTIVATE(Y)
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> ACTIVATE(Z)
2NDSNEG(s(N), cons(X, n_{_cons}(Y, Z))) -> ACTIVATE(Y)
2NDSNEG(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, n_{_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)
ACTIVATE(n_{_cons}(X1, X2)) -> CONS(activate(X1), X2)
ACTIVATE(n_{_cons}(X1, X2)) -> ACTIVATE(X1)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_cons}(X1, X2)) -> ACTIVATE(X1)
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, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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)

three 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''))
ACTIVATE(n_{_from}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))

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_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
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)
ACTIVATE(n_{_cons}(X1, X2)) -> ACTIVATE(X1)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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)

five new Dependency Pairs are created:

ACTIVATE(n_{_s}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))
ACTIVATE(n_{_s}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
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'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))

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_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X'''')))) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X'''')))) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_s}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_s}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))
ACTIVATE(n_{_from}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))
ACTIVATE(n_{_from}(n_{_from}(X''))) -> ACTIVATE(n_{_from}(X''))
ACTIVATE(n_{_cons}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_from}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(X1, X2)) -> ACTIVATE(X1)

nine new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

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:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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''))

three 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'''')))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))

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_{_s}(n_{_from}(n_{_s}(X'''')))) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X'''')))) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_s}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_s}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))
ACTIVATE(n_{_from}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(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_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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''))

five 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_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

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_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_from}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X'''')))) -> ACTIVATE(n_{_from}(n_{_from}(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_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_s}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
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_{_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, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))

nine new Dependency Pairs are created:

ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))

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_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X'''')))) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X'''')))) -> ACTIVATE(n_{_from}(n_{_from}(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_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_s}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
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'''')))
ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X'''')))) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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''))

five 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_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(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''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X'''')))) -> ACTIVATE(n_{_from}(n_{_from}(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_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_s}(n_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_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_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(X1'', X2''))) -> ACTIVATE(n_{_cons}(X1'', X2''))

nine new Dependency Pairs are created:

ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X'''')))) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> 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, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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'''')))

three 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''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
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_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_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_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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'''')))

five 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_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
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'''''''')))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 11

                 ↳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_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))) -> ACTIVATE(n_{_from}(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}(n_{_from}(n_{_from}(X'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_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_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))

nine new Dependency Pairs are created:

ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 12

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
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_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))
ACTIVATE(n_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_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, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_cons}(X1'', X2''), X2)) -> ACTIVATE(n_{_cons}(X1'', X2''))

nine new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 13

                 ↳Forward Instantiation Transformation

Dependency Pairs:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_from}(n_{_from}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_from}(X'''')))

three new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 14

                 ↳Forward Instantiation Transformation

Dependency Pairs:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_from}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_from}(n_{_s}(X'''')))

five new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 15

                 ↳Forward Instantiation Transformation

Dependency Pairs:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_from}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))

nine new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 16

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
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_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
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''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_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_{_cons}(n_{_from}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_s}(n_{_s}(X'''')), X2)) -> ACTIVATE(n_{_s}(n_{_s}(X'''')))

five new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 17

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
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_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_from}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_s}(n_{_cons}(X1'''', X2'''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))

nine new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 18

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
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_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
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''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_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_{_cons}(n_{_from}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))

three new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 19

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
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_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_from}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))

five new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 20

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
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_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
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''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_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_{_cons}(n_{_from}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))

nine new Dependency Pairs are created:

ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''''', X2'''''''''), X2'''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''''', X2'''''''''), X2'''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''''))), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''''))), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''''', X2''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''''', X2''''''''''''))), X2''''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 21

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''''', X2''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''''', X2''''''''''''))), X2''''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''''))), X2''''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''''))), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''''', X2'''''''''), X2'''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''''', X2'''''''''), X2'''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
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_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_from}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''''', X2''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''''', X2''''''''''''))), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''''))), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''''))), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''''))), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''''', X2'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''''')), X2''''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''''')), X2''''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''''', X2'''''''''), X2'''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''''', X2'''''''''), X2'''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_s}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_s}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_from}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_s}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_s}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(n_{_from}(X''''''))), X2)) -> ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X'''''''')))), X2)) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_s}(X'''''''')))))
ACTIVATE(n_{_cons}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''), X2)) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__cons(x₁, x₂)) = 1 + x₁

POL(n__from(x₁)) = x₁

POL(n__s(x₁)) = x₁

POL(ACTIVATE(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 22

                 ↳Dependency Graph

Dependency Pairs:

ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''''))), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_s}(X''''''''))), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_from}(n_{_from}(X''''''''))), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_s}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_cons}(X1'''''', X2'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_s}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_s}(X'''''''')), X2'''''')))
ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_from}(n_{_from}(X'''''''')), X2'''''')))
ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''''', X2'''''''), X2''''')))
ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
ACTIVATE(n_{_s}(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_from}(n_{_cons}(X1'''''''', X2'''''''')))))
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_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''''', X2''''''))))
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_{_cons}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''''', X2''''''))))
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_{_from}(n_{_s}(n_{_from}(n_{_from}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_from}(X''''''))))
ACTIVATE(n_{_from}(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_from}(n_{_from}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_from}(n_{_cons}(X1'''', X2'''')))
ACTIVATE(n_{_s}(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))) -> ACTIVATE(n_{_cons}(n_{_from}(n_{_from}(X'''''')), X2'''))
ACTIVATE(n_{_from}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))) -> ACTIVATE(n_{_cons}(n_{_cons}(X1'''', X2'''''), X2''''))
ACTIVATE(n_{_s}(n_{_s}(n_{_cons}(X1'''', X2'''')))) -> ACTIVATE(n_{_s}(n_{_cons}(X1'''', X2'''')))
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''''''))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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 23

                 ↳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_{_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}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_from}(n_{_s}(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, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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_{_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''''''))))
ACTIVATE(n_{_s}(n_{_s}(n_{_from}(n_{_s}(X''''''))))) -> ACTIVATE(n_{_s}(n_{_from}(n_{_s}(X''''''))))

There are no usable rules for innermost 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 24

                 ↳Dependency Graph

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_{_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'''')))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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 25

                 ↳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, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
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 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 26

                 ↳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, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:29 minutes

POL(n__cons(x₁, x₂))	= 1 + x₁
POL(n__from(x₁))	= x₁
POL(n__s(x₁))	= x₁
POL(ACTIVATE(x₁))	= 1 + x₁