Term Rewriting System R:
[X, Y, Z, X1, X2]
2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

2ND(cons(X, ncons(Y, Z))) -> ACTIVATE(Y)
FROM(X) -> CONS(X, nfrom(ns(X)))
ACTIVATE(ncons(X1, X2)) -> CONS(activate(X1), X2)
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

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

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

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

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ncons(ncons(X1'', X2''), X2)) -> ACTIVATE(ncons(X1'', X2''))
three new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ncons(nfrom(X''), X2)) -> ACTIVATE(nfrom(X''))
five new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ncons(ns(X''), X2)) -> ACTIVATE(ns(X''))
nine new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

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

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nfrom(ncons(ncons(X1'''', X2''''), X2''))) -> ACTIVATE(ncons(ncons(X1'''', X2''''), X2''))
three new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nfrom(ncons(nfrom(X''''), X2''))) -> ACTIVATE(ncons(nfrom(X''''), X2''))
five new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nfrom(ncons(ns(X''''), X2''))) -> ACTIVATE(ncons(ns(X''''), X2''))
nine new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

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

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ns(ncons(ncons(X1'''', X2''''), X2''))) -> ACTIVATE(ncons(ncons(X1'''', X2''''), X2''))
three new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ns(ncons(nfrom(X''''), X2''))) -> ACTIVATE(ncons(nfrom(X''''), X2''))
five new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ns(ncons(ns(X''''), X2''))) -> ACTIVATE(ncons(ns(X''''), X2''))
nine new Dependency Pairs are created:

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

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

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


Rules:


2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
s(X) -> ns(X)
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ns(nfrom(ncons(ncons(X1'''''', X2''''''), X2'''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2''''''), X2'''')))
three new Dependency Pairs are created:

ACTIVATE(ns(nfrom(ncons(ncons(ncons(X1'''''''', X2'''''''''), X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(ncons(X1'''''''', X2'''''''''), X2'''''''), X2''''')))
ACTIVATE(ns(nfrom(ncons(ncons(nfrom(X''''''''), X2''''''''), X2''''0)))) -> ACTIVATE(nfrom(ncons(ncons(nfrom(X''''''''), X2''''''''), X2''''0)))
ACTIVATE(ns(nfrom(ncons(ncons(ns(X''''''''), X2''''''''), X2''''0)))) -> ACTIVATE(nfrom(ncons(ncons(ns(X''''''''), X2''''''''), X2''''0)))

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(ns(ns(nfrom(ncons(ns(X''''''''), X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(ns(X''''''''), X2''''''))))
ACTIVATE(ns(ns(nfrom(ncons(ncons(X1'''''''', X2''''''''), X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(ncons(X1'''''''', X2''''''''), X2''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ncons(ns(X''''''), X2'''')))) -> ACTIVATE(ns(ncons(ns(X''''''), X2'''')))
ACTIVATE(nfrom(ns(nfrom(ncons(ns(X''''''''), X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(ns(X''''''''), X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(ns(X''''''''''), X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(ns(X''''''''''), X2'''''''')))))
ACTIVATE(nfrom(ns(nfrom(ncons(nfrom(X''''''''), X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(nfrom(X''''''''), X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(nfrom(X''''''''''), X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(nfrom(X''''''''''), X2'''''''')))))
ACTIVATE(nfrom(ns(nfrom(ncons(ncons(X1'''''''', X2''''''''), X2''''''))))) -> ACTIVATE(ns(nfrom(ncons(ncons(X1'''''''', X2''''''''), X2''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(ncons(ncons(X1'''''''''', X2''''''''''), X2'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ncons(ncons(X1'''''''''', X2''''''''''), 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(nfrom(ncons(ns(nfrom(ncons(ns(X''''''''''), X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(ns(X''''''''''), X2''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(nfrom(X''''''''''), X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(nfrom(X''''''''''), X2''''''''))), X2'''))
ACTIVATE(nfrom(ncons(ns(nfrom(ncons(ncons(X1'''''''''', X2''''''''''), X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(ncons(X1'''''''''', X2''''''''''), X2''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ns(X''''''), X2'''')))) -> ACTIVATE(nfrom(ncons(ns(X''''''), X2'''')))
ACTIVATE(ncons(ns(nfrom(ncons(ns(X''''''''), X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(ns(X''''''''), X2''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ncons(ns(X''''''''''), X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(ns(X''''''''''), X2''''''''))), X2'''))
ACTIVATE(ncons(ns(nfrom(ncons(nfrom(X''''''''), X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(nfrom(X''''''''), X2''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ncons(nfrom(X''''''''''), X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(nfrom(X''''''''''), X2''''''''))), X2'''))
ACTIVATE(ns(nfrom(ncons(ncons(ns(X''''''''), X2''''''''), X2''''0)))) -> ACTIVATE(nfrom(ncons(ncons(ns(X''''''''), X2''''''''), X2''''0)))
ACTIVATE(ns(nfrom(ncons(ncons(nfrom(X''''''''), X2''''''''), X2''''0)))) -> ACTIVATE(nfrom(ncons(ncons(nfrom(X''''''''), X2''''''''), X2''''0)))
ACTIVATE(ns(nfrom(ncons(ncons(ncons(X1'''''''', X2'''''''''), X2'''''''), X2''''')))) -> ACTIVATE(nfrom(ncons(ncons(ncons(X1'''''''', X2'''''''''), X2'''''''), X2''''')))
ACTIVATE(ncons(ns(nfrom(ncons(ncons(X1'''''''', X2''''''''), X2''''''))), X2)) -> ACTIVATE(ns(nfrom(ncons(ncons(X1'''''''', X2''''''''), X2''''''))))
ACTIVATE(ns(ncons(ns(nfrom(ncons(ncons(X1'''''''''', X2''''''''''), X2''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ncons(ncons(X1'''''''''', X2''''''''''), X2''''''''))), X2'''))
ACTIVATE(ns(ncons(ns(nfrom(ns(X''''''''))), X2'''))) -> ACTIVATE(ncons(ns(nfrom(ns(X''''''''))), X2'''))
ACTIVATE(nfrom(ns(ncons(ns(X''''''), X2'''')))) -> ACTIVATE(ns(ncons(ns(X''''''), X2'''')))
ACTIVATE(ns(nfrom(ns(ncons(ns(X''''''''), X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(ns(X''''''''), X2''''''))))
ACTIVATE(nfrom(ns(ncons(nfrom(X''''''), X2'''')))) -> ACTIVATE(ns(ncons(nfrom(X''''''), X2'''')))
ACTIVATE(ns(nfrom(ns(ncons(nfrom(X''''''''), X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(nfrom(X''''''''), X2''''''))))
ACTIVATE(ns(nfrom(ns(ncons(ncons(X1'''''''', X2''''''''), X2''''''))))) -> ACTIVATE(nfrom(ns(ncons(ncons(X1'''''''', X2''''''''), 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(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(nfrom(nfrom(ncons(ns(X''''''), X2'''')))) -> ACTIVATE(nfrom(ncons(ns(X''''''), X2'''')))
ACTIVATE(ns(nfrom(nfrom(ncons(ns(X''''''''), X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(ns(X''''''''), X2''''''))))
ACTIVATE(nfrom(nfrom(ncons(nfrom(X''''''), X2'''')))) -> ACTIVATE(nfrom(ncons(nfrom(X''''''), X2'''')))
ACTIVATE(ns(nfrom(nfrom(ncons(nfrom(X''''''''), X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(nfrom(X''''''''), X2''''''))))
ACTIVATE(nfrom(nfrom(ncons(ncons(X1'''''', X2''''''), X2'''')))) -> ACTIVATE(nfrom(ncons(ncons(X1'''''', X2''''''), X2'''')))
ACTIVATE(ns(nfrom(nfrom(ncons(ncons(X1'''''''', X2''''''''), X2''''''))))) -> ACTIVATE(nfrom(nfrom(ncons(ncons(X1'''''''', X2''''''''), X2''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
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(ns(X''''''''), X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(ns(X''''''''), X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ncons(nfrom(X''''''''), X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(nfrom(X''''''''), X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ncons(ncons(X1'''''''', X2''''''''), X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(ncons(X1'''''''', X2''''''''), X2'''''')), X2'''))
ACTIVATE(ns(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(ns(X''''''), X2'''')), X2)) -> ACTIVATE(ns(ncons(ns(X''''''), X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(ns(X''''''''), X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(ns(X''''''''), X2'''''')), X2'''))
ACTIVATE(ns(ncons(nfrom(ncons(ns(X''''''''), X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(ns(X''''''''), X2'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(nfrom(X''''''), X2'''')), X2)) -> ACTIVATE(ns(ncons(nfrom(X''''''), X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(nfrom(X''''''''), X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(nfrom(X''''''''), X2'''''')), X2'''))
ACTIVATE(ncons(ns(ncons(ncons(X1'''''', X2''''''), X2'''')), X2)) -> ACTIVATE(ns(ncons(ncons(X1'''''', X2''''''), X2'''')))
ACTIVATE(nfrom(ncons(ns(ncons(ncons(X1'''''''', X2''''''''), X2'''''')), X2'''))) -> ACTIVATE(ncons(ns(ncons(ncons(X1'''''''', X2''''''''), X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(ns(ns(X'''''')), X2'''))) -> ACTIVATE(ncons(ns(ns(X'''''')), X2'''))
ACTIVATE(ncons(nfrom(ncons(ns(X''''''), X2'''')), X2)) -> ACTIVATE(nfrom(ncons(ns(X''''''), X2'''')))
ACTIVATE(nfrom(ncons(nfrom(ncons(ns(X''''''''), X2'''''')), X2'''))) -> ACTIVATE(ncons(nfrom(ncons(ns(X''''''''), X2'''''')), X2'''))
ACTIVATE(nfrom(ncons(nfrom(ncons(n