Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z, X1, X2]
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_from}(X)) -> FROM(activate(X))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(X1, X2)) -> FIRST(activate(X1), activate(X2))

eight new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), X2)) -> FIRST(from(activate(X')), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2)) -> FIRST(s(activate(X')), activate(X2))
ACTIVATE(n_{_first}(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_from}(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2)) -> FIRST(s(activate(X')), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), X2)) -> FIRST(from(activate(X')), activate(X2))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))

13 new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_first}(X1''', X2'''), X2)) -> FIRST(n_{_first}(activate(X1'''), activate(X2''')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2)) -> FIRST(s(activate(X')), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), X2)) -> FIRST(from(activate(X')), activate(X2))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(n_{_from}(X'), X2)) -> FIRST(from(activate(X')), activate(X2))

10 new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(n_{_from}(activate(X'')), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2)) -> FIRST(s(activate(X')), activate(X2))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(n_{_s}(X'), X2)) -> FIRST(s(activate(X')), activate(X2))

nine new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(n_{_s}(activate(X'')), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_from}(X'')), X2)) -> FIRST(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_from}(X'')), X2)) -> FIRST(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(X1', X2)) -> FIRST(X1', activate(X2))

four new Dependency Pairs are created:

ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_from}(X'')), X2)) -> FIRST(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))

13 new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_first}(X1''', X2'), n_{_first}(X1'', X2''))) -> FIRST(first(activate(X1'''), activate(X2')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1'', X2''))) -> FIRST(from(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2'''))) -> FIRST(activate(X1), n_{_first}(activate(X1'''), activate(X2''')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_first}(X1''', X2'), X2''))) -> FIRST(activate(X1), first(first(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_from}(X'), X2''))) -> FIRST(activate(X1), first(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(X1''', activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_from}(X')))) -> FIRST(activate(X1), first(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_from}(X')))) -> FIRST(activate(X1), first(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(X1''', activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_from}(X'), X2''))) -> FIRST(activate(X1), first(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_first}(X1''', X2'), X2''))) -> FIRST(activate(X1), first(first(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1'', X2''))) -> FIRST(from(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2'), n_{_first}(X1'', X2''))) -> FIRST(first(activate(X1'''), activate(X2')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_from}(X'')), X2)) -> FIRST(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(X1, n_{_from}(X'))) -> FIRST(activate(X1), from(activate(X')))

10 new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_from}(X'))) -> FIRST(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_from}(X'))) -> FIRST(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), n_{_from}(activate(X'')))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), from(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_from}(X'')))) -> FIRST(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_s}(X'')))) -> FIRST(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), from(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), from(X''))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_s}(X'')))) -> FIRST(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_from}(X'')))) -> FIRST(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), from(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_from}(X'))) -> FIRST(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_from}(X'))) -> FIRST(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_from}(X')))) -> FIRST(activate(X1), first(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(X1''', activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_from}(X'), X2''))) -> FIRST(activate(X1), first(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_first}(X1''', X2'), X2''))) -> FIRST(activate(X1), first(first(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1'', X2''))) -> FIRST(from(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2'), n_{_first}(X1'', X2''))) -> FIRST(first(activate(X1'''), activate(X2')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_from}(X'')), X2)) -> FIRST(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))

nine new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_s}(X'))) -> FIRST(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), n_{_s}(activate(X'')))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), s(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_from}(X'')))) -> FIRST(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(X''))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_from}(X'')))) -> FIRST(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), s(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_s}(X'))) -> FIRST(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_s}(X'')))) -> FIRST(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_from}(X'')))) -> FIRST(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), from(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_from}(X'))) -> FIRST(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_from}(X'))) -> FIRST(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_from}(X')))) -> FIRST(activate(X1), first(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(X1''', activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_from}(X'), X2''))) -> FIRST(activate(X1), first(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_first}(X1''', X2'), X2''))) -> FIRST(activate(X1), first(first(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1'', X2''))) -> FIRST(from(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2'), n_{_first}(X1'', X2''))) -> FIRST(first(activate(X1'''), activate(X2')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_from}(X'')), X2)) -> FIRST(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), from(X''))

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')

four new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_from}(X'')))) -> FIRST(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), s(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_s}(X'))) -> FIRST(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), from(X''))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_s}(X'')))) -> FIRST(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_from}(X'')))) -> FIRST(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), from(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_from}(X'))) -> FIRST(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_from}(X'))) -> FIRST(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_from}(X')))) -> FIRST(activate(X1), first(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(X1''', activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_from}(X'), X2''))) -> FIRST(activate(X1), first(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_first}(X1''', X2'), X2''))) -> FIRST(activate(X1), first(first(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1'', X2''))) -> FIRST(from(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2'), n_{_first}(X1'', X2''))) -> FIRST(first(activate(X1'''), activate(X2')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_from}(X'')), X2)) -> FIRST(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(X''))

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_first}(n_{_from}(X'), X2')) -> FIRST(from(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_s}(X'))) -> FIRST(from(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X''), n_{_from}(X'))) -> FIRST(from(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1'', X2''))) -> FIRST(from(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(n_{_from}(X'')), X2)) -> FIRST(s(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_s}(X''))) -> FIRST(from(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_from}(X''))) -> FIRST(from(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_from}(X'), n_{_first}(X1', X2''))) -> FIRST(from(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_from}(X''), X2)) -> FIRST(from(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_s}(X'')), X2)) -> FIRST(from(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_from}(X'')), X2)) -> FIRST(from(from(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_from}(n_{_first}(X1', X2'')), X2)) -> FIRST(from(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_from}(X')), X2)) -> FIRST(first(activate(X1''), from(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_from}(X'), X2''), X2)) -> FIRST(first(from(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

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

activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(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(n__from(x₁)) = 1 + x₁

POL(from(x₁)) = 1 + x₁

POL(activate(x₁)) = x₁

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 0

POL(cons(x₁, x₂)) = x₂

POL(FIRST(x₁, x₂)) = x₂

POL(n__s(x₁)) = x₁

POL(nil) = 0

POL(s(x₁)) = x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_from}(X'')))) -> FIRST(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), s(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), from(X''))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_s}(X'')))) -> FIRST(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_from}(X'')))) -> FIRST(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), from(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_from}(X'))) -> FIRST(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_from}(X')))) -> FIRST(activate(X1), first(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(X1''', activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_from}(X'), X2''))) -> FIRST(activate(X1), first(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_first}(X1''', X2'), X2''))) -> FIRST(activate(X1), first(first(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2'), n_{_first}(X1'', X2''))) -> FIRST(first(activate(X1'''), activate(X2')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(X''))

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_from}(X'))) -> FIRST(s(activate(X'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_from}(X''))) -> FIRST(s(activate(X')), from(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_first}(X1', X2''))) -> FIRST(s(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

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

activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(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(n__from(x₁)) = 0

POL(from(x₁)) = 0

POL(activate(x₁)) = x₁

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 0

POL(cons(x₁, x₂)) = x₂

POL(FIRST(x₁, x₂)) = x₂

POL(n__s(x₁)) = 1 + x₁

POL(nil) = 0

POL(s(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳Polynomial Ordering

Dependency Pairs:

ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_from}(X'')))) -> FIRST(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), s(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), from(X''))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_s}(X'')))) -> FIRST(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_from}(X'')))) -> FIRST(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), from(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_from}(X')))) -> FIRST(activate(X1), first(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(X1''', activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_from}(X'), X2''))) -> FIRST(activate(X1), first(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_first}(X1''', X2'), X2''))) -> FIRST(activate(X1), first(first(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2'), n_{_first}(X1'', X2''))) -> FIRST(first(activate(X1'''), activate(X2')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(X''))

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), X2')
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_from}(X'')))) -> FIRST(activate(X1), s(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), s(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), from(X''))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_s}(X'')))) -> FIRST(activate(X1), from(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_from}(X'')))) -> FIRST(activate(X1), from(from(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_from}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), from(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_from}(X''))) -> FIRST(activate(X1), cons(activate(X''), n_{_from}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_from}(X'))) -> FIRST(X1', from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2')), from(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_from}(X')))) -> FIRST(activate(X1), first(activate(X1''), from(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(X1''', activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_from}(X'), X2''))) -> FIRST(activate(X1), first(from(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_first}(X1''', X2'), X2''))) -> FIRST(activate(X1), first(first(activate(X1'''), activate(X2')), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2'), n_{_first}(X1'', X2''))) -> FIRST(first(activate(X1'''), activate(X2')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_from}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), from(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_first}(X1', X2'''))) -> FIRST(first(activate(X1''), activate(X2'')), first(activate(X1'), activate(X2''')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'''), X2)) -> FIRST(first(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_first}(X1', X2''')), X2)) -> FIRST(first(activate(X1''), first(activate(X1'), activate(X2'''))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1''', X2''), X2)) -> FIRST(first(X1''', activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(X''))

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

activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(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(n__from(x₁)) = 0

POL(from(x₁)) = 0

POL(activate(x₁)) = x₁

POL(first(x₁, x₂)) = 1 + x₁ + x₂

POL(0) = 0

POL(cons(x₁, x₂)) = x₂

POL(FIRST(x₁, x₂)) = x₂

POL(n__s(x₁)) = x₁

POL(nil) = 0

POL(s(x₁)) = x₁

POL(ACTIVATE(x₁)) = x₁

POL(n__first(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳Dependency Graph

Dependency Pair:

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:17 minutes

POL(n__from(x₁))	= 1 + x₁
POL(from(x₁))	= 1 + x₁
POL(activate(x₁))	= x₁
POL(first(x₁, x₂))	= x₁ + x₂
POL(0)	= 0
POL(cons(x₁, x₂))	= x₂
POL(FIRST(x₁, x₂))	= x₂
POL(n__s(x₁))	= x₁
POL(nil)	= 0
POL(s(x₁))	= x₁
POL(ACTIVATE(x₁))	= x₁
POL(n__first(x₁, x₂))	= x₁ + x₂