Term Rewriting System R:
[X, Y, Z, X1, X2]
primes -> sieve(from(s(s(0))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(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(s(s(X)), activate(Z)), ncons(Y, nfilter(X, sieve(Y))))
filter(X1, X2) -> nfilter(X1, X2)
sieve(cons(X, Y)) -> cons(X, nfilter(X, sieve(activate(Y))))
cons(X1, X2) -> ncons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nfilter(X1, X2)) -> filter(X1, X2)
activate(ncons(X1, X2)) -> cons(X1, 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

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

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

primes -> sieve(from(s(s(0))))

where the Polynomial interpretation:
 POL(from(x1)) =  2·x1 POL(activate(x1)) =  2·x1 POL(filter(x1, x2)) =  x1 + x2 POL(false) =  0 POL(true) =  0 POL(n__from(x1)) =  x1 POL(n__cons(x1, x2)) =  x1 + x2 POL(if(x1, x2, x3)) =  x1 + 2·x2 + 2·x3 POL(sieve(x1)) =  x1 POL(0) =  0 POL(n__filter(x1, x2)) =  x1 + x2 POL(cons(x1, x2)) =  x1 + x2 POL(primes) =  1 POL(s(x1)) =  x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

cons(X1, X2) -> ncons(X1, X2)
activate(nfilter(X1, X2)) -> filter(X1, X2)
activate(X) -> X
from(X) -> nfrom(X)

where the Polynomial interpretation:
 POL(n__from(x1)) =  x1 POL(from(x1)) =  1 + 2·x1 POL(n__cons(x1, x2)) =  x1 + x2 POL(activate(x1)) =  1 + 2·x1 POL(filter(x1, x2)) =  x1 + x2 POL(if(x1, x2, x3)) =  x1 + 2·x2 + 2·x3 POL(n__filter(x1, x2)) =  x1 + x2 POL(false) =  1 POL(cons(x1, x2)) =  1 + x1 + x2 POL(true) =  1 POL(s(x1)) =  x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS4`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

if(false, X, Y) -> activate(Y)

where the Polynomial interpretation:
 POL(n__from(x1)) =  x1 POL(from(x1)) =  2·x1 POL(n__cons(x1, x2)) =  x1 + x2 POL(activate(x1)) =  2·x1 POL(filter(x1, x2)) =  x1 + x2 POL(if(x1, x2, x3)) =  x1 + 2·x2 + 2·x3 POL(n__filter(x1, x2)) =  x1 + x2 POL(cons(x1, x2)) =  x1 + x2 POL(false) =  1 POL(true) =  0 POL(s(x1)) =  x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

filter(X1, X2) -> nfilter(X1, X2)

where the Polynomial interpretation:
 POL(from(x1)) =  2·x1 POL(n__from(x1)) =  x1 POL(n__cons(x1, x2)) =  x1 + x2 POL(activate(x1)) =  2·x1 POL(filter(x1, x2)) =  1 + x1 + x2 POL(if(x1, x2, x3)) =  x1 + 2·x2 + x3 POL(n__filter(x1, x2)) =  x1 + x2 POL(cons(x1, x2)) =  x1 + x2 POL(true) =  0 POL(s(x1)) =  x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS6`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

from(X) -> cons(X, nfrom(s(X)))

where the Polynomial interpretation:
 POL(n__from(x1)) =  1 + x1 POL(from(x1)) =  2 + 2·x1 POL(n__cons(x1, x2)) =  x1 + x2 POL(activate(x1)) =  2·x1 POL(if(x1, x2, x3)) =  x1 + 2·x2 + x3 POL(cons(x1, x2)) =  x1 + x2 POL(true) =  0 POL(s(x1)) =  x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS7`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

if(true, X, Y) -> activate(X)

where the Polynomial interpretation:
 POL(n__from(x1)) =  x1 POL(from(x1)) =  x1 POL(n__cons(x1, x2)) =  x1 + x2 POL(activate(x1)) =  x1 POL(if(x1, x2, x3)) =  1 + x1 + x2 + x3 POL(cons(x1, x2)) =  x1 + x2 POL(true) =  0
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS8`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

activate(nfrom(X)) -> from(X)

where the Polynomial interpretation:
 POL(n__cons(x1, x2)) =  x1 + x2 POL(n__from(x1)) =  1 + x1 POL(from(x1)) =  x1 POL(activate(x1)) =  x1 POL(cons(x1, x2)) =  x1 + x2
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS9`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

activate(ncons(X1, X2)) -> cons(X1, X2)

where the Polynomial interpretation:
 POL(n__cons(x1, x2)) =  x1 + x2 POL(activate(x1)) =  1 + x1 POL(cons(x1, x2)) =  x1 + x2
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS10`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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