0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
↳5 QDP
↳6 QDPOrderProof (⇔)
↳7 QDP
↳8 QDPOrderProof (⇔)
↳9 QDP
↳10 QDPOrderProof (⇔)
↳11 QDP
↳12 QDPOrderProof (⇔)
↳13 QDP
↳14 PisEmptyProof (⇔)
↳15 TRUE
↳16 QDP
↳17 QDPOrderProof (⇔)
↳18 QDP
↳19 QDPOrderProof (⇔)
↳20 QDP
↳21 QDPOrderProof (⇔)
↳22 QDP
↳23 QDPOrderProof (⇔)
↳24 QDP
↳25 PisEmptyProof (⇔)
↳26 TRUE
↳27 QDP
↳28 QDPOrderProof (⇔)
↳29 QDP
↳30 QDPOrderProof (⇔)
↳31 QDP
↳32 PisEmptyProof (⇔)
↳33 TRUE
↳34 QDP
↳35 QDPOrderProof (⇔)
↳36 QDP
↳37 QDPOrderProof (⇔)
↳38 QDP
↳39 PisEmptyProof (⇔)
↳40 TRUE
↳41 QDP
↳42 QDPOrderProof (⇔)
↳43 QDP
↳44 QDPOrderProof (⇔)
↳45 QDP
↳46 PisEmptyProof (⇔)
↳47 TRUE
↳48 QDP
↳49 QDPOrderProof (⇔)
↳50 QDP
↳51 QDPOrderProof (⇔)
↳52 QDP
↳53 QDPOrderProof (⇔)
↳54 QDP
↳55 QDPOrderProof (⇔)
↳56 QDP
↳57 PisEmptyProof (⇔)
↳58 TRUE
↳59 QDP
↳60 QDPOrderProof (⇔)
↳61 QDP
↳62 QDPOrderProof (⇔)
↳63 QDP
↳64 PisEmptyProof (⇔)
↳65 TRUE
↳66 QDP
↳67 QDPOrderProof (⇔)
↳68 QDP
↳69 QDPOrderProof (⇔)
↳70 QDP
↳71 PisEmptyProof (⇔)
↳72 TRUE
↳73 QDP
↳74 QDPOrderProof (⇔)
↳75 QDP
↳76 QDPOrderProof (⇔)
↳77 QDP
↳78 PisEmptyProof (⇔)
↳79 TRUE
↳80 QDP
↳81 QDPOrderProof (⇔)
↳82 QDP
↳83 QDPOrderProof (⇔)
↳84 QDP
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
ACTIVE(primes) → MARK(sieve(from(s(s(0)))))
ACTIVE(primes) → SIEVE(from(s(s(0))))
ACTIVE(primes) → FROM(s(s(0)))
ACTIVE(primes) → S(s(0))
ACTIVE(primes) → S(0)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
ACTIVE(from(X)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
ACTIVE(head(cons(X, Y))) → MARK(X)
ACTIVE(tail(cons(X, Y))) → MARK(Y)
ACTIVE(if(true, X, Y)) → MARK(X)
ACTIVE(if(false, X, Y)) → MARK(Y)
ACTIVE(filter(s(s(X)), cons(Y, Z))) → MARK(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
ACTIVE(filter(s(s(X)), cons(Y, Z))) → IF(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
ACTIVE(filter(s(s(X)), cons(Y, Z))) → DIVIDES(s(s(X)), Y)
ACTIVE(filter(s(s(X)), cons(Y, Z))) → FILTER(s(s(X)), Z)
ACTIVE(filter(s(s(X)), cons(Y, Z))) → CONS(Y, filter(X, sieve(Y)))
ACTIVE(filter(s(s(X)), cons(Y, Z))) → FILTER(X, sieve(Y))
ACTIVE(filter(s(s(X)), cons(Y, Z))) → SIEVE(Y)
ACTIVE(sieve(cons(X, Y))) → MARK(cons(X, filter(X, sieve(Y))))
ACTIVE(sieve(cons(X, Y))) → CONS(X, filter(X, sieve(Y)))
ACTIVE(sieve(cons(X, Y))) → FILTER(X, sieve(Y))
ACTIVE(sieve(cons(X, Y))) → SIEVE(Y)
MARK(primes) → ACTIVE(primes)
MARK(sieve(X)) → ACTIVE(sieve(mark(X)))
MARK(sieve(X)) → SIEVE(mark(X))
MARK(sieve(X)) → MARK(X)
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(from(X)) → FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(0) → ACTIVE(0)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(head(X)) → HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
MARK(if(X1, X2, X3)) → IF(mark(X1), X2, X3)
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(true) → ACTIVE(true)
MARK(false) → ACTIVE(false)
MARK(filter(X1, X2)) → ACTIVE(filter(mark(X1), mark(X2)))
MARK(filter(X1, X2)) → FILTER(mark(X1), mark(X2))
MARK(filter(X1, X2)) → MARK(X1)
MARK(filter(X1, X2)) → MARK(X2)
MARK(divides(X1, X2)) → ACTIVE(divides(mark(X1), mark(X2)))
MARK(divides(X1, X2)) → DIVIDES(mark(X1), mark(X2))
MARK(divides(X1, X2)) → MARK(X1)
MARK(divides(X1, X2)) → MARK(X2)
SIEVE(mark(X)) → SIEVE(X)
SIEVE(active(X)) → SIEVE(X)
FROM(mark(X)) → FROM(X)
FROM(active(X)) → FROM(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
HEAD(mark(X)) → HEAD(X)
HEAD(active(X)) → HEAD(X)
TAIL(mark(X)) → TAIL(X)
TAIL(active(X)) → TAIL(X)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)
FILTER(mark(X1), X2) → FILTER(X1, X2)
FILTER(X1, mark(X2)) → FILTER(X1, X2)
FILTER(active(X1), X2) → FILTER(X1, X2)
FILTER(X1, active(X2)) → FILTER(X1, X2)
DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
DIVIDES(X1, mark(X2)) → DIVIDES(X1, X2)
DIVIDES(active(X1), X2) → DIVIDES(X1, X2)
DIVIDES(X1, active(X2)) → DIVIDES(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
DIVIDES(X1, mark(X2)) → DIVIDES(X1, X2)
DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
DIVIDES(active(X1), X2) → DIVIDES(X1, X2)
DIVIDES(X1, active(X2)) → DIVIDES(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
DIVIDES(X1, active(X2)) → DIVIDES(X1, X2)
trivial
DIVIDES1: multiset
active1: multiset
DIVIDES(X1, mark(X2)) → DIVIDES(X1, X2)
DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
DIVIDES(active(X1), X2) → DIVIDES(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
DIVIDES(active(X1), X2) → DIVIDES(X1, X2)
trivial
DIVIDES1: multiset
active1: multiset
DIVIDES(X1, mark(X2)) → DIVIDES(X1, X2)
DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
DIVIDES(X1, mark(X2)) → DIVIDES(X1, X2)
mark1 > DIVIDES1
DIVIDES1: multiset
mark1: [1]
DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
[DIVIDES2, mark1]
DIVIDES2: [2,1]
mark1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
FILTER(X1, mark(X2)) → FILTER(X1, X2)
FILTER(mark(X1), X2) → FILTER(X1, X2)
FILTER(active(X1), X2) → FILTER(X1, X2)
FILTER(X1, active(X2)) → FILTER(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FILTER(X1, active(X2)) → FILTER(X1, X2)
trivial
FILTER1: multiset
active1: multiset
FILTER(X1, mark(X2)) → FILTER(X1, X2)
FILTER(mark(X1), X2) → FILTER(X1, X2)
FILTER(active(X1), X2) → FILTER(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FILTER(active(X1), X2) → FILTER(X1, X2)
trivial
FILTER1: multiset
active1: multiset
FILTER(X1, mark(X2)) → FILTER(X1, X2)
FILTER(mark(X1), X2) → FILTER(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FILTER(X1, mark(X2)) → FILTER(X1, X2)
mark1 > FILTER1
FILTER1: multiset
mark1: [1]
FILTER(mark(X1), X2) → FILTER(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FILTER(mark(X1), X2) → FILTER(X1, X2)
[FILTER2, mark1]
FILTER2: [2,1]
mark1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
trivial
mark1: multiset
active1: multiset
IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)
mark1 > IF2
active1 > IF2
IF2: multiset
mark1: multiset
active1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
TAIL(active(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TAIL(active(X)) → TAIL(X)
trivial
active1: multiset
TAIL(mark(X)) → TAIL(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TAIL(mark(X)) → TAIL(X)
trivial
mark1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
HEAD(active(X)) → HEAD(X)
HEAD(mark(X)) → HEAD(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
HEAD(active(X)) → HEAD(X)
trivial
active1: multiset
HEAD(mark(X)) → HEAD(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
HEAD(mark(X)) → HEAD(X)
trivial
mark1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS(X1, active(X2)) → CONS(X1, X2)
trivial
CONS1: multiset
active1: multiset
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS(active(X1), X2) → CONS(X1, X2)
trivial
CONS1: multiset
active1: multiset
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS(X1, mark(X2)) → CONS(X1, X2)
mark1 > CONS1
CONS1: multiset
mark1: [1]
CONS(mark(X1), X2) → CONS(X1, X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS(mark(X1), X2) → CONS(X1, X2)
[CONS2, mark1]
CONS2: [2,1]
mark1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
S(active(X)) → S(X)
S(mark(X)) → S(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
S(active(X)) → S(X)
trivial
active1: multiset
S(mark(X)) → S(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
S(mark(X)) → S(X)
trivial
mark1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
FROM(active(X)) → FROM(X)
FROM(mark(X)) → FROM(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FROM(active(X)) → FROM(X)
trivial
active1: multiset
FROM(mark(X)) → FROM(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FROM(mark(X)) → FROM(X)
trivial
mark1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
SIEVE(active(X)) → SIEVE(X)
SIEVE(mark(X)) → SIEVE(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SIEVE(active(X)) → SIEVE(X)
trivial
active1: multiset
SIEVE(mark(X)) → SIEVE(X)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SIEVE(mark(X)) → SIEVE(X)
trivial
mark1: multiset
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
MARK(sieve(X)) → ACTIVE(sieve(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sieve(X)) → MARK(X)
MARK(primes) → ACTIVE(primes)
ACTIVE(primes) → MARK(sieve(from(s(s(0)))))
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(head(cons(X, Y))) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(tail(cons(X, Y))) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(if(true, X, Y)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(if(false, X, Y)) → MARK(Y)
MARK(head(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
ACTIVE(filter(s(s(X)), cons(Y, Z))) → MARK(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
MARK(tail(X)) → MARK(X)
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
ACTIVE(sieve(cons(X, Y))) → MARK(cons(X, filter(X, sieve(Y))))
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(filter(X1, X2)) → ACTIVE(filter(mark(X1), mark(X2)))
MARK(filter(X1, X2)) → MARK(X1)
MARK(filter(X1, X2)) → MARK(X2)
MARK(divides(X1, X2)) → ACTIVE(divides(mark(X1), mark(X2)))
MARK(divides(X1, X2)) → MARK(X1)
MARK(divides(X1, X2)) → MARK(X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(divides(X1, X2)) → ACTIVE(divides(mark(X1), mark(X2)))
true > [MARK, sieve, mark, from, cons, primes, head, tail, if, filter] > 0 > [s, divides]
true > [MARK, sieve, mark, from, cons, primes, head, tail, if, filter] > false > [s, divides]
MARK: multiset
sieve: multiset
mark: multiset
from: multiset
cons: multiset
s: multiset
primes: multiset
0: multiset
head: multiset
tail: multiset
if: multiset
true: multiset
false: multiset
filter: multiset
divides: multiset
sieve(active(X)) → sieve(X)
sieve(mark(X)) → sieve(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
from(active(X)) → from(X)
from(mark(X)) → from(X)
cons(X1, mark(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(active(X)) → head(X)
head(mark(X)) → head(X)
tail(active(X)) → tail(X)
tail(mark(X)) → tail(X)
divides(X1, mark(X2)) → divides(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(mark(X1), X2) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
MARK(sieve(X)) → ACTIVE(sieve(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sieve(X)) → MARK(X)
MARK(primes) → ACTIVE(primes)
ACTIVE(primes) → MARK(sieve(from(s(s(0)))))
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(head(cons(X, Y))) → MARK(X)
MARK(from(X)) → MARK(X)
ACTIVE(tail(cons(X, Y))) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(if(true, X, Y)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(if(false, X, Y)) → MARK(Y)
MARK(head(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
ACTIVE(filter(s(s(X)), cons(Y, Z))) → MARK(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
MARK(tail(X)) → MARK(X)
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
ACTIVE(sieve(cons(X, Y))) → MARK(cons(X, filter(X, sieve(Y))))
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(filter(X1, X2)) → ACTIVE(filter(mark(X1), mark(X2)))
MARK(filter(X1, X2)) → MARK(X1)
MARK(filter(X1, X2)) → MARK(X2)
MARK(divides(X1, X2)) → MARK(X1)
MARK(divides(X1, X2)) → MARK(X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
[MARK, sieve, mark1, from, primes, head, tail, if, filter] > cons > [s1, 0, divides2, active1]
[MARK, sieve, mark1, from, primes, head, tail, if, filter] > true > [s1, 0, divides2, active1]
[MARK, sieve, mark1, from, primes, head, tail, if, filter] > false > [s1, 0, divides2, active1]
MARK: []
sieve: []
mark1: [1]
from: []
cons: []
s1: multiset
primes: multiset
0: multiset
head: []
tail: []
if: []
true: multiset
false: multiset
filter: []
divides2: multiset
active1: multiset
sieve(active(X)) → sieve(X)
sieve(mark(X)) → sieve(X)
from(active(X)) → from(X)
from(mark(X)) → from(X)
cons(X1, mark(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(active(X)) → head(X)
head(mark(X)) → head(X)
tail(active(X)) → tail(X)
tail(mark(X)) → tail(X)
filter(X1, mark(X2)) → filter(X1, X2)
filter(mark(X1), X2) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
MARK(sieve(X)) → ACTIVE(sieve(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sieve(X)) → MARK(X)
MARK(primes) → ACTIVE(primes)
ACTIVE(primes) → MARK(sieve(from(s(s(0)))))
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(head(cons(X, Y))) → MARK(X)
MARK(from(X)) → MARK(X)
ACTIVE(tail(cons(X, Y))) → MARK(Y)
MARK(s(X)) → MARK(X)
ACTIVE(if(true, X, Y)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(if(false, X, Y)) → MARK(Y)
MARK(head(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
ACTIVE(filter(s(s(X)), cons(Y, Z))) → MARK(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
MARK(tail(X)) → MARK(X)
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
ACTIVE(sieve(cons(X, Y))) → MARK(cons(X, filter(X, sieve(Y))))
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(filter(X1, X2)) → ACTIVE(filter(mark(X1), mark(X2)))
MARK(filter(X1, X2)) → MARK(X1)
MARK(filter(X1, X2)) → MARK(X2)
MARK(divides(X1, X2)) → MARK(X1)
MARK(divides(X1, X2)) → MARK(X2)
active(primes) → mark(sieve(from(s(s(0)))))
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, Y))) → mark(X)
active(tail(cons(X, Y))) → mark(Y)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(filter(s(s(X)), cons(Y, Z))) → mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
active(sieve(cons(X, Y))) → mark(cons(X, filter(X, sieve(Y))))
mark(primes) → active(primes)
mark(sieve(X)) → active(sieve(mark(X)))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
mark(true) → active(true)
mark(false) → active(false)
mark(filter(X1, X2)) → active(filter(mark(X1), mark(X2)))
mark(divides(X1, X2)) → active(divides(mark(X1), mark(X2)))
sieve(mark(X)) → sieve(X)
sieve(active(X)) → sieve(X)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)
filter(mark(X1), X2) → filter(X1, X2)
filter(X1, mark(X2)) → filter(X1, X2)
filter(active(X1), X2) → filter(X1, X2)
filter(X1, active(X2)) → filter(X1, X2)
divides(mark(X1), X2) → divides(X1, X2)
divides(X1, mark(X2)) → divides(X1, X2)
divides(active(X1), X2) → divides(X1, X2)
divides(X1, active(X2)) → divides(X1, X2)