Termination Proof using AProVE

Term Rewriting System R:
[N, X, Y, Z, X1, X2]
terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

TERMS(N) -> SQR(N)
SQR(s(X)) -> S(add(sqr(X), dbl(X)))
SQR(s(X)) -> ADD(sqr(X), dbl(X))
SQR(s(X)) -> SQR(X)
SQR(s(X)) -> DBL(X)
DBL(s(X)) -> S(s(dbl(X)))
DBL(s(X)) -> S(dbl(X))
DBL(s(X)) -> DBL(X)
ADD(s(X), Y) -> S(add(X, Y))
ADD(s(X), Y) -> ADD(X, Y)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
HALF(s(s(X))) -> S(half(X))
HALF(s(s(X))) -> HALF(X)
ACTIVATE(n_{_terms}(X)) -> TERMS(activate(X))
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)

Furthermore, R contains five SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

Dependency Pair:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

The following dependency pair can be strictly oriented:

ADD(s(X), Y) -> ADD(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(s(x₁)) = 1 + x₁

POL(ADD(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 6

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

Dependency Pair:

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

Dependency Pair:

DBL(s(X)) -> DBL(X)

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

The following dependency pair can be strictly oriented:

DBL(s(X)) -> DBL(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(s(x₁)) = 1 + x₁

POL(DBL(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 7

             ↳Dependency Graph

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

Dependency Pair:

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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

         ↳Polynomial Ordering

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

Dependency Pair:

HALF(s(s(X))) -> HALF(X)

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

The following dependency pair can be strictly oriented:

HALF(s(s(X))) -> HALF(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(HALF(x₁)) = 1 + x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 8

             ↳Dependency Graph

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

Dependency Pair:

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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

         ↳Polo

       →DP Problem 4

         ↳Polynomial Ordering

       →DP Problem 5

         ↳Nar

Dependency Pair:

SQR(s(X)) -> SQR(X)

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

The following dependency pair can be strictly oriented:

SQR(s(X)) -> SQR(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(s(x₁)) = 1 + x₁

POL(SQR(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 9

             ↳Dependency Graph

       →DP Problem 5

         ↳Nar

Dependency Pair:

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_first}(X1, X2)) -> FIRST(activate(X1), activate(X2))

eight new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_first}(n_{_terms}(X'), X2)) -> FIRST(terms(activate(X')), activate(X2))

10 new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_terms}(X''), X2)) -> FIRST(cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(X''), X2)) -> FIRST(n_{_terms}(activate(X'')), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_terms}(X'')), X2)) -> FIRST(terms(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_s}(X'')), X2)) -> FIRST(terms(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_first}(X1', X2'')), X2)) -> FIRST(terms(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(X''), X2)) -> FIRST(terms(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_terms}(X''))) -> FIRST(terms(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_s}(X''))) -> FIRST(terms(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_first}(X1', X2''))) -> FIRST(terms(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), X2')) -> FIRST(terms(activate(X')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_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_{_terms}(X'')), X2)) -> FIRST(s(terms(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}(n_{_first}(X1', X2'')), X2)) -> FIRST(s(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_terms}(X''))) -> FIRST(s(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_s}(X''))) -> FIRST(s(activate(X')), s(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(activate(X')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_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_{_terms}(X'), X2''), X2)) -> FIRST(first(terms(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}(n_{_first}(X1', X2'''), X2''), X2)) -> FIRST(first(first(activate(X1'), activate(X2''')), 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_{_terms}(X')), X2)) -> FIRST(first(activate(X1''), terms(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'', 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(activate(X1''), X2'''), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(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''), 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''), activate(X2'')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 13

                 ↳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_{_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_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(activate(X')))
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'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_terms}(X')), X2)) -> FIRST(first(activate(X1''), terms(activate(X'))), 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}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_terms}(X'), X2''), X2)) -> FIRST(first(terms(activate(X')), activate(X2'')), 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_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_terms}(X''))) -> FIRST(s(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(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_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_terms}(X'')), X2)) -> FIRST(s(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(X'), X2')) -> FIRST(terms(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_first}(X1', X2''))) -> FIRST(terms(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_s}(X''))) -> FIRST(terms(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_terms}(X''))) -> FIRST(terms(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X''), X2)) -> FIRST(terms(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_first}(X1', X2'')), X2)) -> FIRST(terms(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_s}(X'')), X2)) -> FIRST(terms(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_terms}(X'')), X2)) -> FIRST(terms(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_terms}(X'))) -> FIRST(activate(X1), terms(activate(X')))
ACTIVATE(n_{_first}(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_first}(X1', X2)) -> FIRST(X1', activate(X2))

four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_first}(X1, n_{_terms}(X'))) -> FIRST(activate(X1), terms(activate(X')))

10 new Dependency Pairs are created:

ACTIVATE(n_{_first}(n_{_terms}(X''), n_{_terms}(X'))) -> FIRST(terms(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_terms}(X'))) -> FIRST(s(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2')), terms(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), n_{_terms}(activate(X'')))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_terms}(X'')))) -> FIRST(activate(X1), terms(terms(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_s}(X'')))) -> FIRST(activate(X1), terms(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), terms(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), terms(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 15

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), terms(X''))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), terms(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_s}(X'')))) -> FIRST(activate(X1), terms(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_terms}(X'')))) -> FIRST(activate(X1), terms(terms(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_terms}(X'))) -> FIRST(s(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_terms}(X''), n_{_terms}(X'))) -> FIRST(terms(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), 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_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(activate(X')))
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'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_terms}(X')), X2)) -> FIRST(first(activate(X1''), terms(activate(X'))), 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}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_terms}(X'), X2''), X2)) -> FIRST(first(terms(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), 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_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_terms}(X''))) -> FIRST(s(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(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_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_terms}(X'')), X2)) -> FIRST(s(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(X'), X2')) -> FIRST(terms(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_first}(X1', X2''))) -> FIRST(terms(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_s}(X''))) -> FIRST(terms(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_terms}(X''))) -> FIRST(terms(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X''), X2)) -> FIRST(terms(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_first}(X1', X2'')), X2)) -> FIRST(terms(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_s}(X'')), X2)) -> FIRST(terms(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_terms}(X'')), X2)) -> FIRST(terms(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_first}(X1, n_{_s}(X'))) -> FIRST(activate(X1), s(activate(X')))

nine new Dependency Pairs are created:

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

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 16

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(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_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_terms}(X'')))) -> FIRST(activate(X1), s(terms(activate(X''))))
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}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_terms}(X''), n_{_s}(X'))) -> FIRST(terms(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), terms(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_s}(X'')))) -> FIRST(activate(X1), terms(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_terms}(X'')))) -> FIRST(activate(X1), terms(terms(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_terms}(X'))) -> FIRST(s(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_terms}(X''), n_{_terms}(X'))) -> FIRST(terms(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), 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_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(activate(X')))
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'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_terms}(X')), X2)) -> FIRST(first(activate(X1''), terms(activate(X'))), 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}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_terms}(X'), X2''), X2)) -> FIRST(first(terms(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), 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_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_terms}(X''))) -> FIRST(s(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(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_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_terms}(X'')), X2)) -> FIRST(s(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(X'), X2')) -> FIRST(terms(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_first}(X1', X2''))) -> FIRST(terms(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_s}(X''))) -> FIRST(terms(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_terms}(X''))) -> FIRST(terms(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X''), X2)) -> FIRST(terms(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_first}(X1', X2'')), X2)) -> FIRST(terms(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_s}(X'')), X2)) -> FIRST(terms(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_terms}(X'')), X2)) -> FIRST(terms(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), terms(X''))

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_first}(X1, n_{_first}(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))

13 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 17

                 ↳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_{_first}(X1''', X2')))) -> FIRST(activate(X1), first(activate(X1''), first(activate(X1'''), activate(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_{_terms}(X')))) -> FIRST(activate(X1), first(activate(X1''), terms(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(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_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_terms}(X'), X2''))) -> FIRST(activate(X1), first(terms(activate(X')), 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_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_first}(X1'', X2''))) -> FIRST(terms(activate(X')), first(activate(X1''), activate(X2'')))
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_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_terms}(X'')))) -> FIRST(activate(X1), s(terms(activate(X''))))
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}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_terms}(X''), n_{_s}(X'))) -> FIRST(terms(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), terms(X''))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), terms(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_s}(X'')))) -> FIRST(activate(X1), terms(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_terms}(X'')))) -> FIRST(activate(X1), terms(terms(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_terms}(X'))) -> FIRST(s(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_terms}(X''), n_{_terms}(X'))) -> FIRST(terms(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), 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_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(activate(X')))
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'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_terms}(X')), X2)) -> FIRST(first(activate(X1''), terms(activate(X'))), 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}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_terms}(X'), X2''), X2)) -> FIRST(first(terms(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), 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_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_terms}(X''))) -> FIRST(s(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(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_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_terms}(X'')), X2)) -> FIRST(s(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(X'), X2')) -> FIRST(terms(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_first}(X1', X2''))) -> FIRST(terms(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_s}(X''))) -> FIRST(terms(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_terms}(X''))) -> FIRST(terms(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X''), X2)) -> FIRST(terms(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_first}(X1', X2'')), X2)) -> FIRST(terms(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_s}(X'')), X2)) -> FIRST(terms(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_terms}(X'')), X2)) -> FIRST(terms(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(X''))

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(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_{_first}(X1, X2')) -> FIRST(activate(X1), X2')

four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 18

                 ↳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}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_terms}(X'), X2')) -> FIRST(terms(activate(X')), 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_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_terms}(X')))) -> FIRST(activate(X1), first(activate(X1''), terms(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(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_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_terms}(X'), X2''))) -> FIRST(activate(X1), first(terms(activate(X')), 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_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_first}(X1'', X2''))) -> FIRST(terms(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(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_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_terms}(X'')))) -> FIRST(activate(X1), s(terms(activate(X''))))
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}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_terms}(X''), n_{_s}(X'))) -> FIRST(terms(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), terms(X''))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), terms(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_s}(X'')))) -> FIRST(activate(X1), terms(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_terms}(X'')))) -> FIRST(activate(X1), terms(terms(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_terms}(X'))) -> FIRST(s(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_terms}(X''), n_{_terms}(X'))) -> FIRST(terms(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), 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_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(activate(X')))
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'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(X1'', n_{_terms}(X')), X2)) -> FIRST(first(activate(X1''), terms(activate(X'))), 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}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_first}(n_{_terms}(X'), X2''), X2)) -> FIRST(first(terms(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), 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_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_terms}(X''))) -> FIRST(s(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(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_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(n_{_terms}(X'')), X2)) -> FIRST(s(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(X'), X2')) -> FIRST(terms(activate(X')), X2')
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_first}(X1', X2''))) -> FIRST(terms(activate(X')), first(activate(X1'), activate(X2'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_s}(X''))) -> FIRST(terms(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X'), n_{_terms}(X''))) -> FIRST(terms(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_terms}(X''), X2)) -> FIRST(terms(X''), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_first}(X1', X2'')), X2)) -> FIRST(terms(first(activate(X1'), activate(X2''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_s}(X'')), X2)) -> FIRST(terms(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(n_{_terms}(n_{_terms}(X'')), X2)) -> FIRST(terms(terms(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_terms}(X)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

The following dependency pairs can be strictly oriented:

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

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

activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
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)
terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(activate(x₁)) = x₁

POL(n__terms(x₁)) = 1 + x₁

POL(sqr(x₁)) = 0

POL(n__s(x₁)) = x₁

POL(terms(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

POL(add(x₁, x₂)) = 0

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 0

POL(cons(x₁, x₂)) = x₂

POL(FIRST(x₁, x₂)) = x₂

POL(dbl(x₁)) = 0

POL(nil) = 0

POL(s(x₁)) = x₁

POL(recip(x₁)) = 0

POL(n__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 19

                 ↳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}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), 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_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_terms}(X')))) -> FIRST(activate(X1), first(activate(X1''), terms(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(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_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_terms}(X'), X2''))) -> FIRST(activate(X1), first(terms(activate(X')), 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_{_s}(X'), n_{_first}(X1'', X2''))) -> FIRST(s(activate(X')), first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_s}(X''))) -> FIRST(activate(X1), s(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_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_terms}(X'')))) -> FIRST(activate(X1), s(terms(activate(X''))))
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}(n_{_s}(X''), n_{_s}(X'))) -> FIRST(s(activate(X'')), s(activate(X')))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), terms(X''))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), terms(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_s}(X'')))) -> FIRST(activate(X1), terms(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_terms}(X'')))) -> FIRST(activate(X1), terms(terms(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2')), terms(activate(X')))
ACTIVATE(n_{_first}(n_{_s}(X''), n_{_terms}(X'))) -> FIRST(s(activate(X'')), terms(activate(X')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), 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_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(activate(X')))
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'', n_{_s}(X')), X2)) -> FIRST(first(activate(X1''), s(activate(X'))), 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}(n_{_first}(n_{_s}(X'), X2''), X2)) -> FIRST(first(s(activate(X')), activate(X2'')), activate(X2))
ACTIVATE(n_{_first}(n_{_s}(X'), X2')) -> FIRST(s(activate(X')), 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_{_s}(X''))) -> FIRST(s(activate(X')), s(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X'), n_{_terms}(X''))) -> FIRST(s(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(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_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), activate(X2))
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_first}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
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_{_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_{_terms}(X'))) -> FIRST(s(activate(X'')), terms(activate(X')))
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_{_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_{_terms}(X''))) -> FIRST(s(activate(X')), terms(activate(X'')))
ACTIVATE(n_{_first}(n_{_s}(X''), X2)) -> FIRST(s(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_{_s}(X'')), X2)) -> FIRST(s(s(activate(X''))), 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_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
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)
terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(activate(x₁)) = x₁

POL(n__terms(x₁)) = 0

POL(sqr(x₁)) = 0

POL(n__s(x₁)) = 1 + x₁

POL(terms(x₁)) = 0

POL(ACTIVATE(x₁)) = x₁

POL(add(x₁, x₂)) = 0

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 0

POL(cons(x₁, x₂)) = x₂

POL(FIRST(x₁, x₂)) = x₂

POL(dbl(x₁)) = 0

POL(nil) = 0

POL(s(x₁)) = 1 + x₁

POL(recip(x₁)) = 0

POL(n__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 20

                 ↳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_{_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_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_terms}(X')))) -> FIRST(activate(X1), first(activate(X1''), terms(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(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_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_terms}(X'), X2''))) -> FIRST(activate(X1), first(terms(activate(X')), 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, n_{_s}(X''))) -> FIRST(activate(X1), s(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_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_terms}(X'')))) -> FIRST(activate(X1), s(terms(activate(X''))))
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_{_terms}(X''))) -> FIRST(activate(X1), terms(X''))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), terms(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_s}(X'')))) -> FIRST(activate(X1), terms(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_terms}(X'')))) -> FIRST(activate(X1), terms(terms(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2')), terms(activate(X')))
ACTIVATE(n_{_first}(X1', X2')) -> FIRST(X1', X2')
ACTIVATE(n_{_first}(X1', n_{_first}(X1'', X2''))) -> FIRST(X1', first(activate(X1''), activate(X2'')))
ACTIVATE(n_{_first}(X1', n_{_s}(X'))) -> FIRST(X1', s(activate(X')))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), X2')) -> FIRST(first(activate(X1''), activate(X2'')), 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_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(activate(X')))
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_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
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_{_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_{_s}(X')))) -> FIRST(activate(X1), first(activate(X1''), s(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1'', n_{_terms}(X')))) -> FIRST(activate(X1), first(activate(X1''), terms(activate(X'))))
ACTIVATE(n_{_first}(X1, n_{_first}(X1''', X2''))) -> FIRST(activate(X1), first(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_{_s}(X'), X2''))) -> FIRST(activate(X1), first(s(activate(X')), activate(X2'')))
ACTIVATE(n_{_first}(X1, n_{_first}(n_{_terms}(X'), X2''))) -> FIRST(activate(X1), first(terms(activate(X')), 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, n_{_s}(X''))) -> FIRST(activate(X1), s(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_{_s}(X'')))) -> FIRST(activate(X1), s(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_s}(n_{_terms}(X'')))) -> FIRST(activate(X1), s(terms(activate(X''))))
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_{_terms}(X''))) -> FIRST(activate(X1), terms(X''))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_first}(X1'', X2')))) -> FIRST(activate(X1), terms(first(activate(X1''), activate(X2'))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_s}(X'')))) -> FIRST(activate(X1), terms(s(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(n_{_terms}(X'')))) -> FIRST(activate(X1), terms(terms(activate(X''))))
ACTIVATE(n_{_first}(X1, n_{_terms}(X''))) -> FIRST(activate(X1), cons(recip(sqr(activate(X''))), n_{_terms}(n_{_s}(activate(X'')))))
ACTIVATE(n_{_first}(X1', n_{_terms}(X'))) -> FIRST(X1', terms(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2'), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2')), terms(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''), n_{_s}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), s(activate(X')))
ACTIVATE(n_{_first}(n_{_first}(X1'', X2''), n_{_terms}(X'))) -> FIRST(first(activate(X1''), activate(X2'')), terms(activate(X')))
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_{_first}(X1'', X2'''))) -> FIRST(activate(X1), first(activate(X1''), X2'''))

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

activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
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)
terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(activate(x₁)) = x₁

POL(n__terms(x₁)) = 0

POL(sqr(x₁)) = 0

POL(n__s(x₁)) = x₁

POL(terms(x₁)) = 0

POL(ACTIVATE(x₁)) = x₁

POL(add(x₁, x₂)) = 0

POL(first(x₁, x₂)) = 1 + x₁ + x₂

POL(0) = 0

POL(cons(x₁, x₂)) = x₂

POL(FIRST(x₁, x₂)) = x₂

POL(dbl(x₁)) = 0

POL(nil) = 0

POL(s(x₁)) = x₁

POL(recip(x₁)) = 0

POL(n__first(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 21

                 ↳Dependency Graph

Dependency Pair:

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

Rules:

terms(N) -> cons(recip(sqr(N)), n_{_terms}(n_{_s}(N)))
terms(X) -> n_{_terms}(X)
sqr(0) -> 0
sqr(s(X)) -> s(add(sqr(X), dbl(X)))
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
half(0) -> 0
half(s(0)) -> 0
half(s(s(X))) -> s(half(X))
half(dbl(X)) -> X
s(X) -> n_{_s}(X)
activate(n_{_terms}(X)) -> terms(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_first}(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:18 minutes

POL(activate(x₁))	= x₁
POL(n__terms(x₁))	= 1 + x₁
POL(sqr(x₁))	= 0
POL(n__s(x₁))	= x₁
POL(terms(x₁))	= 1 + x₁
POL(ACTIVATE(x₁))	= x₁
POL(add(x₁, x₂))	= 0
POL(first(x₁, x₂))	= x₁ + x₂
POL(0)	= 0
POL(cons(x₁, x₂))	= x₂
POL(FIRST(x₁, x₂))	= x₂
POL(dbl(x₁))	= 0
POL(nil)	= 0
POL(s(x₁))	= x₁
POL(recip(x₁))	= 0
POL(n__first(x₁, x₂))	= x₁ + x₂