Term Rewriting System R:
[X, Y, Z, X1, X2]
primes -> sieve(from(s(s(0))))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
head(cons(X, Y)) -> X
tail(cons(X, Y)) -> activate(Y)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), nfilter(ns(ns(X)), activate(Z)), ncons(Y, nfilter(X, nsieve(Y))))
filter(X1, X2) -> nfilter(X1, X2)
sieve(cons(X, Y)) -> cons(X, nfilter(X, nsieve(activate(Y))))
sieve(X) -> nsieve(X)
s(X) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nfilter(X1, X2)) -> filter(activate(X1), activate(X2))
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nsieve(X)) -> sieve(activate(X))
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

head(cons(X, Y)) -> X
tail(cons(X, Y)) -> activate(Y)
filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), nfilter(ns(ns(X)), activate(Z)), ncons(Y, nfilter(X, nsieve(Y))))
sieve(cons(X, Y)) -> cons(X, nfilter(X, nsieve(activate(Y))))


   R
     ↳RRRI
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

PRIMES -> SIEVE(from(s(s(0))))
PRIMES -> FROM(s(s(0)))
PRIMES -> S(s(0))
PRIMES -> S(0)
FROM(X) -> CONS(X, nfrom(ns(X)))
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfilter(X1, X2)) -> FILTER(activate(X1), activate(X2))
ACTIVATE(nfilter(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfilter(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(ncons(X1, X2)) -> CONS(activate(X1), X2)
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nsieve(X)) -> SIEVE(activate(X))
ACTIVATE(nsieve(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Size-Change Principle


Dependency Pairs:

ACTIVATE(nsieve(X)) -> ACTIVATE(X)
ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfilter(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfilter(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)


Rules:


primes -> sieve(from(s(s(0))))
sieve(X) -> nsieve(X)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
cons(X1, X2) -> ncons(X1, X2)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nfilter(X1, X2)) -> filter(activate(X1), activate(X2))
activate(ncons(X1, X2)) -> cons(activate(X1), X2)
activate(nsieve(X)) -> sieve(activate(X))
activate(X) -> X
filter(X1, X2) -> nfilter(X1, X2)





We number the DPs as follows:
  1. ACTIVATE(nsieve(X)) -> ACTIVATE(X)
  2. ACTIVATE(ncons(X1, X2)) -> ACTIVATE(X1)
  3. ACTIVATE(nfilter(X1, X2)) -> ACTIVATE(X2)
  4. ACTIVATE(nfilter(X1, X2)) -> ACTIVATE(X1)
  5. ACTIVATE(ns(X)) -> ACTIVATE(X)
  6. ACTIVATE(nfrom(X)) -> ACTIVATE(X)
and get the following Size-Change Graph(s):
{6, 5, 4, 3, 2, 1} , {6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{6, 5, 4, 3, 2, 1} , {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:
ncons(x1, x2) -> ncons(x1, x2)
nfrom(x1) -> nfrom(x1)
nfilter(x1, x2) -> nfilter(x1, x2)
nsieve(x1) -> nsieve(x1)
ns(x1) -> ns(x1)

We obtain no new DP problems.

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