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

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

sqr(s(X)) -> s(nadd(nsqr(activate(X)), ndbl(activate(X))))
dbl(s(X)) -> s(ns(ndbl(activate(X))))
add(s(X), Y) -> s(nadd(activate(X), Y))
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(activate(X), activate(Z)))


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

TERMS(N) -> SQR(N)
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
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Size-Change Principle


Dependency Pairs:

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)


Rules:


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





We number the DPs as follows:
  1. ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
  2. ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
  3. ACTIVATE(ndbl(X)) -> ACTIVATE(X)
  4. ACTIVATE(nsqr(X)) -> ACTIVATE(X)
  5. ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
  6. ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
  7. ACTIVATE(nterms(X)) -> ACTIVATE(X)
and get the following Size-Change Graph(s):
{7, 6, 5, 4, 3, 2, 1} , {7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{7, 6, 5, 4, 3, 2, 1} , {7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
nterms(x1) -> nterms(x1)
nsqr(x1) -> nsqr(x1)
nadd(x1, x2) -> nadd(x1, x2)
ndbl(x1) -> ndbl(x1)
nfirst(x1, x2) -> nfirst(x1, x2)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes