Term Rewriting System R:
[N, X, Y, X1, X2, Z]
terms(N) -> cons(recip(sqr(N)), nterms(ns(N)))
terms(X) -> nterms(X)
sqr(0) -> 0
sqr(s(X)) -> s(nadd(nsqr(activate(X)), ndbl(activate(X))))
sqr(X) -> nsqr(X)
dbl(0) -> 0
dbl(s(X)) -> s(ns(ndbl(activate(X))))
dbl(X) -> ndbl(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
s(X) -> ns(X)
activate(nterms(X)) -> terms(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nsqr(X)) -> sqr(activate(X))
activate(ndbl(X)) -> dbl(activate(X))
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

TERMS(N) -> SQR(N)
SQR(s(X)) -> S(nadd(nsqr(activate(X)), ndbl(activate(X))))
SQR(s(X)) -> ACTIVATE(X)
DBL(s(X)) -> S(ns(ndbl(activate(X))))
DBL(s(X)) -> ACTIVATE(X)
ADD(s(X), Y) -> S(nadd(activate(X), Y))
ADD(s(X), Y) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nterms(X)) -> TERMS(activate(X))
ACTIVATE(nterms(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(X)
ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nsqr(X)) -> SQR(activate(X))
ACTIVATE(nsqr(X)) -> ACTIVATE(X)
ACTIVATE(ndbl(X)) -> DBL(activate(X))
ACTIVATE(ndbl(X)) -> ACTIVATE(X)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(ndbl(X)) -> ACTIVATE(X)
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(ndbl(X)) -> DBL(activate(X))
ACTIVATE(nsqr(X)) -> ACTIVATE(X)
ACTIVATE(nsqr(X)) -> SQR(activate(X))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(nterms(X)) -> ACTIVATE(X)
ACTIVATE(nterms(X)) -> TERMS(activate(X))
SQR(s(X)) -> ACTIVATE(X)
TERMS(N) -> SQR(N)


Rules:


terms(N) -> cons(recip(sqr(N)), nterms(ns(N)))
terms(X) -> nterms(X)
sqr(0) -> 0
sqr(s(X)) -> s(nadd(nsqr(activate(X)), ndbl(activate(X))))
sqr(X) -> nsqr(X)
dbl(0) -> 0
dbl(s(X)) -> s(ns(ndbl(activate(X))))
dbl(X) -> ndbl(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
s(X) -> ns(X)
activate(nterms(X)) -> terms(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nsqr(X)) -> sqr(activate(X))
activate(ndbl(X)) -> dbl(activate(X))
activate(nfirst(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(nadd(X1, X2)) -> ADD(activate(X1), activate(X2))
14 new Dependency Pairs are created:

ACTIVATE(nadd(nterms(X'), X2)) -> ADD(terms(activate(X')), activate(X2))
ACTIVATE(nadd(ns(X'), X2)) -> ADD(s(X'), activate(X2))
ACTIVATE(nadd(nadd(X1'', X2''), X2)) -> ADD(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(nsqr(X'), X2)) -> ADD(sqr(activate(X')), activate(X2))
ACTIVATE(nadd(ndbl(X'), X2)) -> ADD(dbl(activate(X')), activate(X2))
ACTIVATE(nadd(nfirst(X1'', X2''), X2)) -> ADD(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(X1', X2)) -> ADD(X1', activate(X2))
ACTIVATE(nadd(X1, nterms(X'))) -> ADD(activate(X1), terms(activate(X')))
ACTIVATE(nadd(X1, ns(X'))) -> ADD(activate(X1), s(X'))
ACTIVATE(nadd(X1, nadd(X1'', X2''))) -> ADD(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, nsqr(X'))) -> ADD(activate(X1), sqr(activate(X')))
ACTIVATE(nadd(X1, ndbl(X'))) -> ADD(activate(X1), dbl(activate(X')))
ACTIVATE(nadd(X1, nfirst(X1'', X2''))) -> ADD(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, X2')) -> ADD(activate(X1), X2')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

ACTIVATE(nadd(X1, X2')) -> ADD(activate(X1), X2')
ACTIVATE(nadd(X1, nfirst(X1'', X2''))) -> ADD(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, ndbl(X'))) -> ADD(activate(X1), dbl(activate(X')))
ACTIVATE(nadd(X1, nsqr(X'))) -> ADD(activate(X1), sqr(activate(X')))
ACTIVATE(nadd(X1, nadd(X1'', X2''))) -> ADD(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, ns(X'))) -> ADD(activate(X1), s(X'))
ACTIVATE(nadd(X1, nterms(X'))) -> ADD(activate(X1), terms(activate(X')))
ACTIVATE(nadd(X1', X2)) -> ADD(X1', activate(X2))
ACTIVATE(nadd(nfirst(X1'', X2''), X2)) -> ADD(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(ndbl(X'), X2)) -> ADD(dbl(activate(X')), activate(X2))
ACTIVATE(nadd(nsqr(X'), X2)) -> ADD(sqr(activate(X')), activate(X2))
ACTIVATE(nadd(nadd(X1'', X2''), X2)) -> ADD(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(ns(X'), X2)) -> ADD(s(X'), activate(X2))
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(nadd(nterms(X'), X2)) -> ADD(terms(activate(X')), activate(X2))
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(ndbl(X)) -> ACTIVATE(X)
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(ndbl(X)) -> DBL(activate(X))
ACTIVATE(nsqr(X)) -> ACTIVATE(X)
ACTIVATE(nsqr(X)) -> SQR(activate(X))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nterms(X)) -> ACTIVATE(X)
SQR(s(X)) -> ACTIVATE(X)
TERMS(N) -> SQR(N)
ACTIVATE(nterms(X)) -> TERMS(activate(X))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)


Rules:


terms(N) -> cons(recip(sqr(N)), nterms(ns(N)))
terms(X) -> nterms(X)
sqr(0) -> 0
sqr(s(X)) -> s(nadd(nsqr(activate(X)), ndbl(activate(X))))
sqr(X) -> nsqr(X)
dbl(0) -> 0
dbl(s(X)) -> s(ns(ndbl(activate(X))))
dbl(X) -> ndbl(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
s(X) -> ns(X)
activate(nterms(X)) -> terms(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nsqr(X)) -> sqr(activate(X))
activate(ndbl(X)) -> dbl(activate(X))
activate(nfirst(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(nsqr(X)) -> SQR(activate(X))
seven new Dependency Pairs are created:

ACTIVATE(nsqr(nterms(X''))) -> SQR(terms(activate(X'')))
ACTIVATE(nsqr(ns(X''))) -> SQR(s(X''))
ACTIVATE(nsqr(nadd(X1', X2'))) -> SQR(add(activate(X1'), activate(X2')))
ACTIVATE(nsqr(nsqr(X''))) -> SQR(sqr(activate(X'')))
ACTIVATE(nsqr(ndbl(X''))) -> SQR(dbl(activate(X'')))
ACTIVATE(nsqr(nfirst(X1', X2'))) -> SQR(first(activate(X1'), activate(X2')))
ACTIVATE(nsqr(X'')) -> SQR(X'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nsqr(X'')) -> SQR(X'')
ACTIVATE(nsqr(nfirst(X1', X2'))) -> SQR(first(activate(X1'), activate(X2')))
ACTIVATE(nsqr(ndbl(X''))) -> SQR(dbl(activate(X'')))
ACTIVATE(nsqr(nsqr(X''))) -> SQR(sqr(activate(X'')))
ACTIVATE(nsqr(nadd(X1', X2'))) -> SQR(add(activate(X1'), activate(X2')))
ACTIVATE(nsqr(ns(X''))) -> SQR(s(X''))
ACTIVATE(nsqr(nterms(X''))) -> SQR(terms(activate(X'')))
ACTIVATE(nadd(X1, nfirst(X1'', X2''))) -> ADD(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, ndbl(X'))) -> ADD(activate(X1), dbl(activate(X')))
ACTIVATE(nadd(X1, nsqr(X'))) -> ADD(activate(X1), sqr(activate(X')))
ACTIVATE(nadd(X1, nadd(X1'', X2''))) -> ADD(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, ns(X'))) -> ADD(activate(X1), s(X'))
ACTIVATE(nadd(X1, nterms(X'))) -> ADD(activate(X1), terms(activate(X')))
ACTIVATE(nadd(X1', X2)) -> ADD(X1', activate(X2))
ACTIVATE(nadd(nfirst(X1'', X2''), X2)) -> ADD(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(ndbl(X'), X2)) -> ADD(dbl(activate(X')), activate(X2))
ACTIVATE(nadd(nsqr(X'), X2)) -> ADD(sqr(activate(X')), activate(X2))
ACTIVATE(nadd(nadd(X1'', X2''), X2)) -> ADD(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(ns(X'), X2)) -> ADD(s(X'), activate(X2))
ACTIVATE(nadd(nterms(X'), X2)) -> ADD(terms(activate(X')), activate(X2))
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(ndbl(X)) -> ACTIVATE(X)
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(ndbl(X)) -> DBL(activate(X))
ACTIVATE(nsqr(X)) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nterms(X)) -> ACTIVATE(X)
SQR(s(X)) -> ACTIVATE(X)
TERMS(N) -> SQR(N)
ACTIVATE(nterms(X)) -> TERMS(activate(X))
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2')) -> ADD(activate(X1), X2')


Rules:


terms(N) -> cons(recip(sqr(N)), nterms(ns(N)))
terms(X) -> nterms(X)
sqr(0) -> 0
sqr(s(X)) -> s(nadd(nsqr(activate(X)), ndbl(activate(X))))
sqr(X) -> nsqr(X)
dbl(0) -> 0
dbl(s(X)) -> s(ns(ndbl(activate(X))))
dbl(X) -> ndbl(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
s(X) -> ns(X)
activate(nterms(X)) -> terms(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nsqr(X)) -> sqr(activate(X))
activate(ndbl(X)) -> dbl(activate(X))
activate(nfirst(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(ndbl(X)) -> DBL(activate(X))
seven new Dependency Pairs are created:

ACTIVATE(ndbl(nterms(X''))) -> DBL(terms(activate(X'')))
ACTIVATE(ndbl(ns(X''))) -> DBL(s(X''))
ACTIVATE(ndbl(nadd(X1', X2'))) -> DBL(add(activate(X1'), activate(X2')))
ACTIVATE(ndbl(nsqr(X''))) -> DBL(sqr(activate(X'')))
ACTIVATE(ndbl(ndbl(X''))) -> DBL(dbl(activate(X'')))
ACTIVATE(ndbl(nfirst(X1', X2'))) -> DBL(first(activate(X1'), activate(X2')))
ACTIVATE(ndbl(X'')) -> DBL(X'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pairs:

ACTIVATE(ndbl(X'')) -> DBL(X'')
ACTIVATE(ndbl(nfirst(X1', X2'))) -> DBL(first(activate(X1'), activate(X2')))
ACTIVATE(ndbl(ndbl(X''))) -> DBL(dbl(activate(X'')))
ACTIVATE(ndbl(nsqr(X''))) -> DBL(sqr(activate(X'')))
ACTIVATE(ndbl(nadd(X1', X2'))) -> DBL(add(activate(X1'), activate(X2')))
ACTIVATE(ndbl(ns(X''))) -> DBL(s(X''))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(ndbl(nterms(X''))) -> DBL(terms(activate(X'')))
ACTIVATE(nsqr(X'')) -> SQR(X'')
ACTIVATE(nsqr(nfirst(X1', X2'))) -> SQR(first(activate(X1'), activate(X2')))
ACTIVATE(nsqr(ndbl(X''))) -> SQR(dbl(activate(X'')))
ACTIVATE(nsqr(nsqr(X''))) -> SQR(sqr(activate(X'')))
ACTIVATE(nsqr(nadd(X1', X2'))) -> SQR(add(activate(X1'), activate(X2')))
ACTIVATE(nsqr(ns(X''))) -> SQR(s(X''))
ACTIVATE(nsqr(nterms(X''))) -> SQR(terms(activate(X'')))
ACTIVATE(nadd(X1, X2')) -> ADD(activate(X1), X2')
ACTIVATE(nadd(X1, nfirst(X1'', X2''))) -> ADD(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, ndbl(X'))) -> ADD(activate(X1), dbl(activate(X')))
ACTIVATE(nadd(X1, nsqr(X'))) -> ADD(activate(X1), sqr(activate(X')))
ACTIVATE(nadd(X1, nadd(X1'', X2''))) -> ADD(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, ns(X'))) -> ADD(activate(X1), s(X'))
ACTIVATE(nadd(X1, nterms(X'))) -> ADD(activate(X1), terms(activate(X')))
ACTIVATE(nadd(X1', X2)) -> ADD(X1', activate(X2))
ACTIVATE(nadd(nfirst(X1'', X2''), X2)) -> ADD(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(ndbl(X'), X2)) -> ADD(dbl(activate(X')), activate(X2))
ACTIVATE(nadd(nsqr(X'), X2)) -> ADD(sqr(activate(X')), activate(X2))
ACTIVATE(nadd(nadd(X1'', X2''), X2)) -> ADD(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(ns(X'), X2)) -> ADD(s(X'), activate(X2))
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(nadd(nterms(X'), X2)) -> ADD(terms(activate(X')), activate(X2))
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(ndbl(X)) -> ACTIVATE(X)
ACTIVATE(nsqr(X)) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nterms(X)) -> ACTIVATE(X)
SQR(s(X)) -> ACTIVATE(X)
TERMS(N) -> SQR(N)
ACTIVATE(nterms(X)) -> TERMS(activate(X))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)


Rules:


terms(N) -> cons(recip(sqr(N)), nterms(ns(N)))
terms(X) -> nterms(X)
sqr(0) -> 0
sqr(s(X)) -> s(nadd(nsqr(activate(X)), ndbl(activate(X))))
sqr(X) -> nsqr(X)
dbl(0) -> 0
dbl(s(X)) -> s(ns(ndbl(activate(X))))
dbl(X) -> ndbl(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
s(X) -> ns(X)
activate(nterms(X)) -> terms(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nsqr(X)) -> sqr(activate(X))
activate(ndbl(X)) -> dbl(activate(X))
activate(nfirst(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(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
14 new Dependency Pairs are created:

ACTIVATE(nfirst(nterms(X'), X2)) -> FIRST(terms(activate(X')), activate(X2))
ACTIVATE(nfirst(ns(X'), X2)) -> FIRST(s(X'), activate(X2))
ACTIVATE(nfirst(nadd(X1'', X2''), X2)) -> FIRST(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nfirst(nsqr(X'), X2)) -> FIRST(sqr(activate(X')), activate(X2))
ACTIVATE(nfirst(ndbl(X'), X2)) -> FIRST(dbl(activate(X')), activate(X2))
ACTIVATE(nfirst(nfirst(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nfirst(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(nfirst(X1, nterms(X'))) -> FIRST(activate(X1), terms(activate(X')))
ACTIVATE(nfirst(X1, ns(X'))) -> FIRST(activate(X1), s(X'))
ACTIVATE(nfirst(X1, nadd(X1'', X2''))) -> FIRST(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(nfirst(X1, nsqr(X'))) -> FIRST(activate(X1), sqr(activate(X')))
ACTIVATE(nfirst(X1, ndbl(X'))) -> FIRST(activate(X1), dbl(activate(X')))
ACTIVATE(nfirst(X1, nfirst(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nfirst(X1, X2')) -> FIRST(activate(X1), X2')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

ACTIVATE(nfirst(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(nfirst(X1, nfirst(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nfirst(X1, ndbl(X'))) -> FIRST(activate(X1), dbl(activate(X')))
ACTIVATE(nfirst(X1, nsqr(X'))) -> FIRST(activate(X1), sqr(activate(X')))
ACTIVATE(nfirst(X1, nadd(X1'', X2''))) -> FIRST(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(nfirst(X1, ns(X'))) -> FIRST(activate(X1), s(X'))
ACTIVATE(nfirst(X1, nterms(X'))) -> FIRST(activate(X1), terms(activate(X')))
ACTIVATE(nfirst(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(nfirst(nfirst(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nfirst(ndbl(X'), X2)) -> FIRST(dbl(activate(X')), activate(X2))
ACTIVATE(nfirst(nsqr(X'), X2)) -> FIRST(sqr(activate(X')), activate(X2))
ACTIVATE(nfirst(nadd(X1'', X2''), X2)) -> FIRST(add(activate(X1''), activate(X2'')), activate(X2))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nfirst(ns(X'), X2)) -> FIRST(s(X'), activate(X2))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(nfirst(nterms(X'), X2)) -> FIRST(terms(activate(X')), activate(X2))
ACTIVATE(ndbl(nfirst(X1', X2'))) -> DBL(first(activate(X1'), activate(X2')))
ACTIVATE(ndbl(ndbl(X''))) -> DBL(dbl(activate(X'')))
ACTIVATE(ndbl(nsqr(X''))) -> DBL(sqr(activate(X'')))
ACTIVATE(ndbl(nadd(X1', X2'))) -> DBL(add(activate(X1'), activate(X2')))
ACTIVATE(ndbl(ns(X''))) -> DBL(s(X''))
ACTIVATE(ndbl(nterms(X''))) -> DBL(terms(activate(X'')))
ACTIVATE(nsqr(X'')) -> SQR(X'')
ACTIVATE(nsqr(nfirst(X1', X2'))) -> SQR(first(activate(X1'), activate(X2')))
ACTIVATE(nsqr(ndbl(X''))) -> SQR(dbl(activate(X'')))
ACTIVATE(nsqr(nsqr(X''))) -> SQR(sqr(activate(X'')))
ACTIVATE(nsqr(nadd(X1', X2'))) -> SQR(add(activate(X1'), activate(X2')))
ACTIVATE(nsqr(ns(X''))) -> SQR(s(X''))
ACTIVATE(nsqr(nterms(X''))) -> SQR(terms(activate(X'')))
ACTIVATE(nadd(X1, X2')) -> ADD(activate(X1), X2')
ACTIVATE(nadd(X1, nfirst(X1'', X2''))) -> ADD(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, ndbl(X'))) -> ADD(activate(X1), dbl(activate(X')))
ACTIVATE(nadd(X1, nsqr(X'))) -> ADD(activate(X1), sqr(activate(X')))
ACTIVATE(nadd(X1, nadd(X1'', X2''))) -> ADD(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(nadd(X1, ns(X'))) -> ADD(activate(X1), s(X'))
ACTIVATE(nadd(X1, nterms(X'))) -> ADD(activate(X1), terms(activate(X')))
ACTIVATE(nadd(X1', X2)) -> ADD(X1', activate(X2))
ACTIVATE(nadd(nfirst(X1'', X2''), X2)) -> ADD(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(ndbl(X'), X2)) -> ADD(dbl(activate(X')), activate(X2))
ACTIVATE(nadd(nsqr(X'), X2)) -> ADD(sqr(activate(X')), activate(X2))
ACTIVATE(nadd(nadd(X1'', X2''), X2)) -> ADD(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nadd(ns(X'), X2)) -> ADD(s(X'), activate(X2))
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(nadd(nterms(X'), X2)) -> ADD(terms(activate(X')), activate(X2))
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ndbl(X)) -> ACTIVATE(X)
ACTIVATE(nsqr(X)) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nterms(X)) -> ACTIVATE(X)
SQR(s(X)) -> ACTIVATE(X)
TERMS(N) -> SQR(N)
ACTIVATE(nterms(X)) -> TERMS(activate(X))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(ndbl(X'')) -> DBL(X'')


Rules:


terms(N) -> cons(recip(sqr(N)), nterms(ns(N)))
terms(X) -> nterms(X)
sqr(0) -> 0
sqr(s(X)) -> s(nadd(nsqr(activate(X)), ndbl(activate(X))))
sqr(X) -> nsqr(X)
dbl(0) -> 0
dbl(s(X)) -> s(ns(ndbl(activate(X))))
dbl(X) -> ndbl(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
s(X) -> ns(X)
activate(nterms(X)) -> terms(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nsqr(X)) -> sqr(activate(X))
activate(ndbl(X)) -> dbl(activate(X))
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(X) -> X




Termination of R could not be shown.
Duration:
0:15 minutes