Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N, Y, YS, X1, X2]
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
MINUS(s(X), s(Y)) -> MINUS(X, Y)
QUOT(s(X), s(Y)) -> S(quot(minus(X, Y), s(Y)))
QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
QUOT(s(X), s(Y)) -> MINUS(X, Y)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> QUOT(X, Y)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_from}(X)) -> FROM(activate(X))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

Dependency Pair:

MINUS(s(X), s(Y)) -> MINUS(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

The following dependency pair can be strictly oriented:

MINUS(s(X), s(Y)) -> MINUS(X, Y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MINUS(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

Dependency Pair:

QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

The following dependency pair can be strictly oriented:

QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUOT(x₁, x₂)) = x₁

POL(0) = 0

POL(minus(x₁, x₂)) = 0

POL(n__s(x₁)) = 0

POL(s(x₁)) = 1

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 6

             ↳Dependency Graph

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))

eight new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2)) -> ZWQUOT(from(activate(X')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2)) -> ZWQUOT(s(activate(X')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2)) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(X1', X2)) -> ZWQUOT(X1', activate(X2))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X'))) -> ZWQUOT(activate(X1), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X'))) -> ZWQUOT(activate(X1), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', X2)) -> ZWQUOT(X1', activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2)) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2)) -> ZWQUOT(s(activate(X')), activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2)) -> ZWQUOT(from(activate(X')), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2)) -> ZWQUOT(from(activate(X')), activate(X2))

10 new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(n_{_from}(activate(X'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X'))) -> ZWQUOT(activate(X1), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', X2)) -> ZWQUOT(X1', activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2)) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2)) -> ZWQUOT(s(activate(X')), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2)) -> ZWQUOT(s(activate(X')), activate(X2))

nine new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(n_{_s}(activate(X'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X'))) -> ZWQUOT(activate(X1), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', X2)) -> ZWQUOT(X1', activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2)) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2)) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), activate(X2))

13 new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2'''), X2)) -> ZWQUOT(n_{_zWquot}(activate(X1'''), activate(X2''')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X'))) -> ZWQUOT(activate(X1), from(activate(X')))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(X1', X2)) -> ZWQUOT(X1', activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(X1', X2)) -> ZWQUOT(X1', activate(X2))

four new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X'))) -> ZWQUOT(activate(X1), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(X1, n_{_from}(X'))) -> ZWQUOT(activate(X1), from(activate(X')))

10 new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_from}(X'))) -> ZWQUOT(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_from}(X'))) -> ZWQUOT(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), n_{_from}(activate(X'')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), from(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), from(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), from(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), from(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_from}(X'))) -> ZWQUOT(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_from}(X'))) -> ZWQUOT(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X'))) -> ZWQUOT(activate(X1), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(X1, n_{_s}(X'))) -> ZWQUOT(activate(X1), s(activate(X')))

nine new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_s}(X'))) -> ZWQUOT(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_s}(X'))) -> ZWQUOT(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X''))) -> ZWQUOT(activate(X1), n_{_s}(activate(X'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), s(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X''))) -> ZWQUOT(activate(X1), s(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(X1, n_{_s}(X''))) -> ZWQUOT(activate(X1), s(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), s(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_s}(X'))) -> ZWQUOT(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_s}(X'))) -> ZWQUOT(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), from(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_from}(X'))) -> ZWQUOT(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_from}(X'))) -> ZWQUOT(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), from(X''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), activate(X2'')))

13 new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(zWquot(activate(X1'''), activate(X2')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1''', X2'''))) -> ZWQUOT(activate(X1), n_{_zWquot}(activate(X1'''), activate(X2''')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_from}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_s}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_zWquot}(X1''', X2'), X2''))) -> ZWQUOT(activate(X1), zWquot(zWquot(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1''', X2''))) -> ZWQUOT(activate(X1), zWquot(X1''', activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_from}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_s}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_zWquot}(X1''', X2')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), zWquot(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2'''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), X2'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2'''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), X2'''))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_zWquot}(X1''', X2')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), zWquot(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_s}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_from}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1''', X2''))) -> ZWQUOT(activate(X1), zWquot(X1''', activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_zWquot}(X1''', X2'), X2''))) -> ZWQUOT(activate(X1), zWquot(zWquot(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_s}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_from}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(zWquot(activate(X1'''), activate(X2')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), s(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_s}(X'))) -> ZWQUOT(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_s}(X'))) -> ZWQUOT(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), from(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), from(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_from}(X'))) -> ZWQUOT(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_from}(X'))) -> ZWQUOT(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X''))) -> ZWQUOT(activate(X1), s(X''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

ACTIVATE(n_{_zWquot}(X1, X2')) -> ZWQUOT(activate(X1), X2')

four new Dependency Pairs are created:

ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 15

                 ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_zWquot}(X1''', X2')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), zWquot(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_s}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_from}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1''', X2''))) -> ZWQUOT(activate(X1), zWquot(X1''', activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_zWquot}(X1''', X2'), X2''))) -> ZWQUOT(activate(X1), zWquot(zWquot(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_s}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_from}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(zWquot(activate(X1'''), activate(X2')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X''))) -> ZWQUOT(activate(X1), s(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), s(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_s}(X'))) -> ZWQUOT(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_s}(X'))) -> ZWQUOT(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), from(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), from(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_from}(X'))) -> ZWQUOT(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_from}(X'))) -> ZWQUOT(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2'''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(activate(x₁)) = x₁

POL(minus(x₁, x₂)) = 0

POL(n__s(x₁)) = x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__from(x₁)) = 1 + x₁

POL(0) = 0

POL(n__zWquot(x₁, x₂)) = x₁ + x₂

POL(zWquot(x₁, x₂)) = x₁ + x₂

POL(cons(x₁, x₂)) = x₂

POL(nil) = 0

POL(quot(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(ZWQUOT(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 16

                 ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_zWquot}(X1''', X2')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), zWquot(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_s}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_from}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1''', X2''))) -> ZWQUOT(activate(X1), zWquot(X1''', activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_zWquot}(X1''', X2'), X2''))) -> ZWQUOT(activate(X1), zWquot(zWquot(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_s}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_from}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(zWquot(activate(X1'''), activate(X2')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X''))) -> ZWQUOT(activate(X1), s(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), s(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_s}(X'))) -> ZWQUOT(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_s}(X'))) -> ZWQUOT(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), from(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), from(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_from}(X'))) -> ZWQUOT(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_from}(X'))) -> ZWQUOT(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2'''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 0

POL(activate(x₁)) = x₁

POL(minus(x₁, x₂)) = 0

POL(n__s(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__from(x₁)) = 0

POL(0) = 0

POL(n__zWquot(x₁, x₂)) = x₁ + x₂

POL(zWquot(x₁, x₂)) = x₁ + x₂

POL(cons(x₁, x₂)) = x₂

POL(nil) = 0

POL(quot(x₁, x₂)) = 0

POL(s(x₁)) = 1 + x₁

POL(ZWQUOT(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 17

                 ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_zWquot}(X1''', X2')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), zWquot(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_s}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_from}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1''', X2''))) -> ZWQUOT(activate(X1), zWquot(X1''', activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_zWquot}(X1''', X2'), X2''))) -> ZWQUOT(activate(X1), zWquot(zWquot(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_s}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_from}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(zWquot(activate(X1'''), activate(X2')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X''))) -> ZWQUOT(activate(X1), s(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), s(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_s}(X'))) -> ZWQUOT(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_s}(X'))) -> ZWQUOT(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), from(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), from(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_from}(X'))) -> ZWQUOT(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_from}(X'))) -> ZWQUOT(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2'''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), X2'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_zWquot}(X1', X2')) -> ZWQUOT(X1', X2')
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2')) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_zWquot}(n_{_s}(X'), X2')) -> ZWQUOT(s(activate(X')), X2')
ACTIVATE(n_{_zWquot}(n_{_from}(X'), X2')) -> ZWQUOT(from(activate(X')), X2')
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_zWquot}(X1''', X2')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), zWquot(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_s}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', n_{_from}(X')))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1''', X2''))) -> ZWQUOT(activate(X1), zWquot(X1''', activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_zWquot}(X1''', X2'), X2''))) -> ZWQUOT(activate(X1), zWquot(zWquot(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_s}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(n_{_from}(X'), X2''))) -> ZWQUOT(activate(X1), zWquot(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1', n_{_zWquot}(X1'', X2''))) -> ZWQUOT(X1', zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(zWquot(activate(X1'''), activate(X2')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1'', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1''), activate(X2'')))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(X''))) -> ZWQUOT(activate(X1), s(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), s(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_s}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1', n_{_s}(X'))) -> ZWQUOT(X1', s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_s}(X'))) -> ZWQUOT(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_s}(X'))) -> ZWQUOT(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), from(X''))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_zWquot}(X1'', X2')))) -> ZWQUOT(activate(X1), from(zWquot(activate(X1''), activate(X2'))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_s}(X'')))) -> ZWQUOT(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(n_{_from}(X'')))) -> ZWQUOT(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_zWquot}(X1, n_{_from}(X''))) -> ZWQUOT(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_zWquot}(X1', n_{_from}(X'))) -> ZWQUOT(X1', from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), n_{_from}(X'))) -> ZWQUOT(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), n_{_from}(X'))) -> ZWQUOT(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_zWquot}(X1', X2'''))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), zWquot(activate(X1'), activate(X2''')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_s}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2''), n_{_from}(X'))) -> ZWQUOT(zWquot(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', X2'''), X2)) -> ZWQUOT(zWquot(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_zWquot}(X1', X2''')), X2)) -> ZWQUOT(zWquot(activate(X1''), zWquot(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_s}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1'', n_{_from}(X')), X2)) -> ZWQUOT(zWquot(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(X1''', X2''), X2)) -> ZWQUOT(zWquot(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_zWquot}(X1', X2'''), X2''), X2)) -> ZWQUOT(zWquot(zWquot(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_s}(X'), X2''), X2)) -> ZWQUOT(zWquot(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_zWquot}(n_{_from}(X'), X2''), X2)) -> ZWQUOT(zWquot(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(s(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_s}(X''))) -> ZWQUOT(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X'), n_{_from}(X''))) -> ZWQUOT(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_s}(X''), X2)) -> ZWQUOT(s(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(s(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_s}(X'')), X2)) -> ZWQUOT(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_s}(n_{_from}(X'')), X2)) -> ZWQUOT(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_zWquot}(X1', X2''))) -> ZWQUOT(from(activate(X')), zWquot(activate(X1'), activate(X2'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_s}(X''))) -> ZWQUOT(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X'), n_{_from}(X''))) -> ZWQUOT(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(from(X''), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_zWquot}(X1', X2'')), X2)) -> ZWQUOT(from(zWquot(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_s}(X'')), X2)) -> ZWQUOT(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(n_{_from}(X'')), X2)) -> ZWQUOT(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_zWquot}(n_{_from}(X''), X2)) -> ZWQUOT(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_zWquot}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_zWquot}(X1, n_{_zWquot}(X1'', X2'''))) -> ZWQUOT(activate(X1), zWquot(activate(X1''), X2'''))

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 0

POL(activate(x₁)) = x₁

POL(minus(x₁, x₂)) = 0

POL(n__s(x₁)) = 0

POL(ACTIVATE(x₁)) = x₁

POL(n__from(x₁)) = 0

POL(0) = 0

POL(n__zWquot(x₁, x₂)) = 1 + x₁ + x₂

POL(zWquot(x₁, x₂)) = 1 + x₁ + x₂

POL(cons(x₁, x₂)) = x₂

POL(nil) = 0

POL(quot(x₁, x₂)) = 0

POL(s(x₁)) = 0

POL(ZWQUOT(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 18

                 ↳Dependency Graph

       →DP Problem 4

         ↳Nar

Dependency Pairs:

ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Narrowing Transformation

Dependency Pair:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

four new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_from}(X''))) -> SEL(N, from(activate(X'')))
SEL(s(N), cons(X, n_{_s}(X''))) -> SEL(N, s(activate(X'')))
SEL(s(N), cons(X, n_{_zWquot}(X1', X2'))) -> SEL(N, zWquot(activate(X1'), activate(X2')))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Narrowing Transformation

Dependency Pairs:

SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, n_{_zWquot}(X1', X2'))) -> SEL(N, zWquot(activate(X1'), activate(X2')))
SEL(s(N), cons(X, n_{_s}(X''))) -> SEL(N, s(activate(X'')))
SEL(s(N), cons(X, n_{_from}(X''))) -> SEL(N, from(activate(X'')))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

SEL(s(N), cons(X, n_{_from}(X''))) -> SEL(N, from(activate(X'')))

six new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, n_{_from}(activate(X''')))
SEL(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> SEL(N, from(from(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> SEL(N, from(s(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(n_{_zWquot}(X1', X2')))) -> SEL(N, from(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, from(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 20

                 ↳Narrowing Transformation

Dependency Pairs:

SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, from(X'''))
SEL(s(N), cons(X, n_{_from}(n_{_zWquot}(X1', X2')))) -> SEL(N, from(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> SEL(N, from(from(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> SEL(N, from(s(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
SEL(s(N), cons(X, n_{_zWquot}(X1', X2'))) -> SEL(N, zWquot(activate(X1'), activate(X2')))
SEL(s(N), cons(X, n_{_s}(X''))) -> SEL(N, s(activate(X'')))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

SEL(s(N), cons(X, n_{_s}(X''))) -> SEL(N, s(activate(X'')))

five new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_s}(X'''))) -> SEL(N, n_{_s}(activate(X''')))
SEL(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> SEL(N, s(from(activate(X'''))))
SEL(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> SEL(N, s(s(activate(X'''))))
SEL(s(N), cons(X, n_{_s}(n_{_zWquot}(X1', X2')))) -> SEL(N, s(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_s}(X'''))) -> SEL(N, s(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 21

                 ↳Narrowing Transformation

Dependency Pairs:

SEL(s(N), cons(X, n_{_s}(X'''))) -> SEL(N, s(X'''))
SEL(s(N), cons(X, n_{_s}(n_{_zWquot}(X1', X2')))) -> SEL(N, s(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> SEL(N, s(s(activate(X'''))))
SEL(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> SEL(N, s(from(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(n_{_zWquot}(X1', X2')))) -> SEL(N, from(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> SEL(N, from(from(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> SEL(N, from(s(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, n_{_zWquot}(X1', X2'))) -> SEL(N, zWquot(activate(X1'), activate(X2')))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, from(X'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

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

SEL(s(N), cons(X, n_{_zWquot}(X1', X2'))) -> SEL(N, zWquot(activate(X1'), activate(X2')))

nine new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_zWquot}(X1'', X2''))) -> SEL(N, n_{_zWquot}(activate(X1''), activate(X2'')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_from}(X''), X2'))) -> SEL(N, zWquot(from(activate(X'')), activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_s}(X''), X2'))) -> SEL(N, zWquot(s(activate(X'')), activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2'))) -> SEL(N, zWquot(zWquot(activate(X1''), activate(X2'')), activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(X1'', X2'))) -> SEL(N, zWquot(X1'', activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_from}(X'')))) -> SEL(N, zWquot(activate(X1'), from(activate(X''))))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_s}(X'')))) -> SEL(N, zWquot(activate(X1'), s(activate(X''))))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_zWquot}(X1'', X2'')))) -> SEL(N, zWquot(activate(X1'), zWquot(activate(X1''), activate(X2''))))
SEL(s(N), cons(X, n_{_zWquot}(X1', X2''))) -> SEL(N, zWquot(activate(X1'), X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 22

                 ↳Polynomial Ordering

Dependency Pairs:

SEL(s(N), cons(X, n_{_zWquot}(X1', X2''))) -> SEL(N, zWquot(activate(X1'), X2''))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_zWquot}(X1'', X2'')))) -> SEL(N, zWquot(activate(X1'), zWquot(activate(X1''), activate(X2''))))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_s}(X'')))) -> SEL(N, zWquot(activate(X1'), s(activate(X''))))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_from}(X'')))) -> SEL(N, zWquot(activate(X1'), from(activate(X''))))
SEL(s(N), cons(X, n_{_zWquot}(X1'', X2'))) -> SEL(N, zWquot(X1'', activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2'))) -> SEL(N, zWquot(zWquot(activate(X1''), activate(X2'')), activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_s}(X''), X2'))) -> SEL(N, zWquot(s(activate(X'')), activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_from}(X''), X2'))) -> SEL(N, zWquot(from(activate(X'')), activate(X2')))
SEL(s(N), cons(X, n_{_s}(n_{_zWquot}(X1', X2')))) -> SEL(N, s(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> SEL(N, s(s(activate(X'''))))
SEL(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> SEL(N, s(from(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, from(X'''))
SEL(s(N), cons(X, n_{_from}(n_{_zWquot}(X1', X2')))) -> SEL(N, from(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> SEL(N, from(from(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> SEL(N, from(s(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, n_{_s}(X'''))) -> SEL(N, s(X'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

The following dependency pairs can be strictly oriented:

SEL(s(N), cons(X, n_{_zWquot}(X1', X2''))) -> SEL(N, zWquot(activate(X1'), X2''))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_zWquot}(X1'', X2'')))) -> SEL(N, zWquot(activate(X1'), zWquot(activate(X1''), activate(X2''))))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_s}(X'')))) -> SEL(N, zWquot(activate(X1'), s(activate(X''))))
SEL(s(N), cons(X, n_{_zWquot}(X1', n_{_from}(X'')))) -> SEL(N, zWquot(activate(X1'), from(activate(X''))))
SEL(s(N), cons(X, n_{_zWquot}(X1'', X2'))) -> SEL(N, zWquot(X1'', activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_zWquot}(X1'', X2''), X2'))) -> SEL(N, zWquot(zWquot(activate(X1''), activate(X2'')), activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_s}(X''), X2'))) -> SEL(N, zWquot(s(activate(X'')), activate(X2')))
SEL(s(N), cons(X, n_{_zWquot}(n_{_from}(X''), X2'))) -> SEL(N, zWquot(from(activate(X'')), activate(X2')))
SEL(s(N), cons(X, n_{_s}(n_{_zWquot}(X1', X2')))) -> SEL(N, s(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> SEL(N, s(s(activate(X'''))))
SEL(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> SEL(N, s(from(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, from(X'''))
SEL(s(N), cons(X, n_{_from}(n_{_zWquot}(X1', X2')))) -> SEL(N, from(zWquot(activate(X1'), activate(X2'))))
SEL(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> SEL(N, from(from(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> SEL(N, from(s(activate(X'''))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, n_{_s}(X'''))) -> SEL(N, s(X'''))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 0

POL(activate(x₁)) = 0

POL(SEL(x₁, x₂)) = x₁

POL(minus(x₁, x₂)) = 0

POL(n__s(x₁)) = 0

POL(n__from(x₁)) = 0

POL(0) = 0

POL(n__zWquot(x₁, x₂)) = 0

POL(zWquot(x₁, x₂)) = 0

POL(cons(x₁, x₂)) = 0

POL(nil) = 0

POL(quot(x₁, x₂)) = 0

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

           →DP Problem 19

             ↳Nar

...

               →DP Problem 23

                 ↳Dependency Graph

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n_{_zWquot}(activate(XS), activate(YS)))
zWquot(X1, X2) -> n_{_zWquot}(X1, X2)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_zWquot}(X1, X2)) -> zWquot(activate(X1), activate(X2))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:34 minutes

POL(QUOT(x₁, x₂))	= x₁
POL(0)	= 0
POL(minus(x₁, x₂))	= 0
POL(n__s(x₁))	= 0
POL(s(x₁))	= 1

POL(from(x₁))	= 1 + x₁
POL(activate(x₁))	= x₁
POL(minus(x₁, x₂))	= 0
POL(n__s(x₁))	= x₁
POL(ACTIVATE(x₁))	= x₁
POL(n__from(x₁))	= 1 + x₁
POL(0)	= 0
POL(n__zWquot(x₁, x₂))	= x₁ + x₂
POL(zWquot(x₁, x₂))	= x₁ + x₂
POL(cons(x₁, x₂))	= x₂
POL(nil)	= 0
POL(quot(x₁, x₂))	= 0
POL(s(x₁))	= x₁
POL(ZWQUOT(x₁, x₂))	= x₁ + x₂

POL(from(x₁))	= 0
POL(activate(x₁))	= 0
POL(SEL(x₁, x₂))	= x₁
POL(minus(x₁, x₂))	= 0
POL(n__s(x₁))	= 0
POL(n__from(x₁))	= 0
POL(0)	= 0
POL(n__zWquot(x₁, x₂))	= 0
POL(zWquot(x₁, x₂))	= 0
POL(cons(x₁, x₂))	= 0
POL(nil)	= 0
POL(quot(x₁, x₂))	= 0
POL(s(x₁))	= 1 + x₁