Term Rewriting System R:
[X, Z, N, Y, X1, X2]
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

FROM(X) -> CONS(X, nfrom(ns(X)))
2NDSPOS(s(N), cons(X, ncons(Y, Z))) -> ACTIVATE(Y)
2NDSPOS(s(N), cons(X, ncons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons(X, ncons(Y, Z))) -> ACTIVATE(Z)
2NDSNEG(s(N), cons(X, ncons(Y, Z))) -> ACTIVATE(Y)
2NDSNEG(s(N), cons(X, ncons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, ncons(Y, Z))) -> ACTIVATE(Z)
PI(X) -> 2NDSPOS(X, from(0))
PI(X) -> FROM(0)
PLUS(s(X), Y) -> S(plus(X, Y))
PLUS(s(X), Y) -> PLUS(X, Y)
TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
TIMES(s(X), Y) -> TIMES(X, Y)
SQUARE(X) -> TIMES(X, X)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(ncons(X1, X2)) -> CONS(activate(X1), X2)
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(nfrom(X)) -> ACTIVATE(X)
three new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(nfrom(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ns(X)) -> ACTIVATE(X)
five new Dependency Pairs are created:

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfrom(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ncons(X1, X2)) -> ACTIVATE(X1)
nine new Dependency Pairs are created:

ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(nfrom(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
three new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(nfrom(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
five new Dependency Pairs are created:

ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(nfrom(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
nine new Dependency Pairs are created:

ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ns(ns(X''))) -> ACTIVATE(ns(X''))
five new Dependency Pairs are created:

ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 8
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ns(ncons(X1'', X2''))) -> ACTIVATE(ncons(X1'', X2''))
nine new Dependency Pairs are created:

ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 9
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
three new Dependency Pairs are created:

ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 10
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
five new Dependency Pairs are created:

ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 11
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ns(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
nine new Dependency Pairs are created:

ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 12
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
nine new Dependency Pairs are created:

ACTIVATE(ncons(ncons(ncons(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ncons(ncons(nfrom(nfrom(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(nfrom(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(ns(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 13
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(ns(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(nfrom(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ncons(ncons(nfrom(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(ncons(nfrom(nfrom(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(ncons(ncons(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ncons(nfrom(nfrom(X'''')), X2)) -> ACTIVATE(nfrom(nfrom(X'''')))
three new Dependency Pairs are created:

ACTIVATE(ncons(nfrom(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(nfrom(nfrom(ns(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ncons(nfrom(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 14
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(ncons(ns(nfrom(ns(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(nfrom(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(nfrom(ns(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ncons(ncons(nfrom(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ncons(nfrom(nfrom(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(ncons(ncons(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ncons(nfrom(ns(X'''')), X2)) -> ACTIVATE(nfrom(ns(X'''')))
five new Dependency Pairs are created:

ACTIVATE(ncons(nfrom(ns(ns(X''''''))), X2)) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(nfrom(ns(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ncons(nfrom(ns(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ncons(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 15
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(nfrom(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ncons(nfrom(ns(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(ns(ns(X''''''))), X2)) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(ncons(nfrom(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(nfrom(ns(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ncons(nfrom(nfrom(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(ncons(ncons(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(ncons(ns(nfrom(ns(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ncons(nfrom(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
nine new Dependency Pairs are created:

ACTIVATE(ncons(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 16
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(ncons(ns(nfrom(ns(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(nfrom(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(nfrom(ns(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(ns(ns(X''''''))), X2)) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(ncons(nfrom(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ncons(nfrom(nfrom(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(ncons(ncons(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ncons(ns(ns(X'''')), X2)) -> ACTIVATE(ns(ns(X'''')))
five new Dependency Pairs are created:

ACTIVATE(ncons(ns(ns(ns(X''''''))), X2)) -> ACTIVATE(ns(ns(ns(X''''''))))
ACTIVATE(ncons(ns(ns(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ns(ns(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ncons(ns(ns(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ncons(ns(ns(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(ncons(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 17
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(nfrom(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ns(ns(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(ns(ns(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ncons(ns(ns(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ns(ns(ns(X''''''))), X2)) -> ACTIVATE(ns(ns(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ncons(nfrom(ns(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(ns(ns(X''''''))), X2)) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(ncons(nfrom(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(nfrom(ns(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ncons(nfrom(nfrom(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(ncons(ncons(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(ncons(ns(nfrom(ns(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ncons(ns(ncons(X1'''', X2'''')), X2)) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
nine new Dependency Pairs are created:

ACTIVATE(ncons(ns(ncons(ncons(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(ns(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(ncons(ns(ncons(nfrom(nfrom(X'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ncons(ns(ncons(nfrom(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(ns(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(ns(ncons(ns(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(ns(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(ns(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 18
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(ncons(ns(nfrom(ns(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(nfrom(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(ns(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ns(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(ns(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(ns(X''''''))), X2)) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(nfrom(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(ns(ncons(nfrom(nfrom(X'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ncons(ns(ncons(ncons(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(ns(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(nfrom(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ns(ns(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(ns(ns(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ns(ns(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(ns(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ncons(ns(ns(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ns(ns(ns(X''''''))), X2)) -> ACTIVATE(ns(ns(ns(X''''''))))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')), X2)) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))) -> ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''''')), X2'''''')))
ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''', X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(nfrom(ns(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(nfrom(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(nfrom(ncons(X1'''', X2'''')))) -> ACTIVATE(nfrom(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(nfrom(ns(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(ncons(nfrom(nfrom(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(nfrom(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ncons(nfrom(ns(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(nfrom(ns(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(nfrom(ns(ns(X''''''))), X2)) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ncons(ncons(nfrom(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ns(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(ns(ncons(X1'''', X2'''')))) -> ACTIVATE(ns(ncons(X1'''', X2'''')))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ncons(nfrom(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ncons(ncons(nfrom(nfrom(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(nfrom(X'''''')), X2'''))
ACTIVATE(ncons(ncons(ncons(X1'''', X2'''''), X2''''), X2)) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(nfrom(ncons(ncons(X1'''', X2'''''), X2''''))) -> ACTIVATE(ncons(ncons(X1'''', X2'''''), X2''''))
ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2'''''''), X2''''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(ncons(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, ncons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, ncons(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) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(ncons(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(ncons(ns(nfrom(nfrom(X''''''))), X2)) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
three new Dependency Pairs are created:

ACTIVATE(ncons(ns(nfrom(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(ns(nfrom(nfrom(nfrom(X'''''''')))))
ACTIVATE(ncons(ns(nfrom(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(ns(nfrom(nfrom(ns(X'''''''')))))
ACTIVATE(ncons(ns(nfrom(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 19
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ncons(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(nfrom(nfrom(X''''''''))), X2'''), X2)) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ncons(ns(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ncons(ns(ns(X'''''')), X2'''), X2)) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''), X2)) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(ns(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(ns(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(ns(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ns(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(ns(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(ns(ns(X'''''''')), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ns(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(ncons(X1'''''', X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(X1'''''', X2''''''))))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(X1'''''''', X2''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ncons(X1'''''''''', X2''''''''''))), X2'''''')))
ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(ns(X''''''''''))), X2'''''')))
ACTIVATE(ncons(ns(nfrom(nfrom(ncons(X1'''''''', X2'''''''')))), X2)) -> ACTIVATE(ns(nfrom(nfrom(ncons(X1'''''''', X2'''''''')))))
ACTIVATE(ncons(ns(nfrom(nfrom(ns(X'''''''')))), X2)) -> ACTIVATE(ns(nfrom(nfrom(ns(X'''''''')))))
ACTIVATE(ncons(ns(nfrom(nfrom(nfrom(X'''''''')))), X2)) -> ACTIVATE(ns(nfrom(nfrom(nfrom(X'''''''')))))
ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(nfrom(X''''''''))), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(nfrom(nfrom(X''''''''''))), X2'''''')))
ACTIVATE(ncons(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(nfrom(ncons(ns(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(X1'''''', X2'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(nfrom(ncons(X1'''''''', X2'''''''')), X2'''''')))
ACTIVATE(ns(ncons(nfrom(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(nfrom(ns(X'''''''')), X2'''''')), X2)) -> ACTIVATE(ns(ncons(nfrom(ns(X'''''''')), X2'''''')))
ACTIVATE(ncons(ns(ncons(nfrom(nfrom(