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)
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(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
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(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