Termination w.r.t. Q proof of /tmp/tmpslTvBf/ExIntrod_GM99

Termination w.r.t. Q of the given QTRS could not be shown:

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 PisEmptyProof (⇔)

    ↳13 TRUE

  ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 PisEmptyProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 PisEmptyProof (⇔)

    ↳25 TRUE

  ↳26 QDP

    ↳27 QDPOrderProof (⇔)

    ↳28 QDP

    ↳29 PisEmptyProof (⇔)

    ↳30 TRUE

  ↳31 QDP

    ↳32 QDPOrderProof (⇔)

    ↳33 QDP

    ↳34 PisEmptyProof (⇔)

    ↳35 TRUE

  ↳36 QDP

    ↳37 QDPOrderProof (⇔)

    ↳38 QDP

    ↳39 QDPOrderProof (⇔)

    ↳40 QDP

    ↳41 PisEmptyProof (⇔)

    ↳42 TRUE

  ↳43 QDP

    ↳44 QDPOrderProof (⇔)

    ↳45 QDP

    ↳46 PisEmptyProof (⇔)

    ↳47 TRUE

  ↳48 QDP

    ↳49 QDPOrderProof (⇔)

    ↳50 QDP

    ↳51 PisEmptyProof (⇔)

    ↳52 TRUE

  ↳53 QDP

    ↳54 QDPOrderProof (⇔)

    ↳55 QDP

    ↳56 PisEmptyProof (⇔)

    ↳57 TRUE

  ↳58 QDP

    ↳59 QDPOrderProof (⇔)

    ↳60 QDP

    ↳61 QDPOrderProof (⇔)

    ↳62 QDP

    ↳63 PisEmptyProof (⇔)

    ↳64 TRUE

  ↳65 QDP

    ↳66 QDPOrderProof (⇔)

    ↳67 QDP

    ↳68 QDPOrderProof (⇔)

    ↳69 QDP

    ↳70 QDPOrderProof (⇔)

    ↳71 QDP

    ↳72 QDPOrderProof (⇔)

    ↳73 QDP

    ↳74 QDPOrderProof (⇔)

    ↳75 QDP

    ↳76 QDPOrderProof (⇔)

    ↳77 QDP

    ↳78 PisEmptyProof (⇔)

    ↳79 TRUE

  ↳80 QDP

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
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))) → CONS(X, filter(X, sieve(Y)))
ACTIVE(sieve(cons(X, Y))) → FILTER(X, sieve(Y))
ACTIVE(sieve(cons(X, Y))) → SIEVE(Y)
ACTIVE(sieve(X)) → SIEVE(active(X))
ACTIVE(sieve(X)) → ACTIVE(X)
ACTIVE(from(X)) → FROM(active(X))
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(head(X)) → HEAD(active(X))
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(tail(X)) → TAIL(active(X))
ACTIVE(tail(X)) → ACTIVE(X)
ACTIVE(if(X1, X2, X3)) → IF(active(X1), X2, X3)
ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(filter(X1, X2)) → FILTER(active(X1), X2)
ACTIVE(filter(X1, X2)) → ACTIVE(X1)
ACTIVE(filter(X1, X2)) → FILTER(X1, active(X2))
ACTIVE(filter(X1, X2)) → ACTIVE(X2)
ACTIVE(divides(X1, X2)) → DIVIDES(active(X1), X2)
ACTIVE(divides(X1, X2)) → ACTIVE(X1)
ACTIVE(divides(X1, X2)) → DIVIDES(X1, active(X2))
ACTIVE(divides(X1, X2)) → ACTIVE(X2)
SIEVE(mark(X)) → SIEVE(X)
FROM(mark(X)) → FROM(X)
S(mark(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
HEAD(mark(X)) → HEAD(X)
TAIL(mark(X)) → TAIL(X)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
FILTER(mark(X1), X2) → FILTER(X1, X2)
FILTER(X1, mark(X2)) → FILTER(X1, X2)
DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
DIVIDES(X1, mark(X2)) → DIVIDES(X1, X2)
PROPER(sieve(X)) → SIEVE(proper(X))
PROPER(sieve(X)) → PROPER(X)
PROPER(from(X)) → FROM(proper(X))
PROPER(from(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(head(X)) → HEAD(proper(X))
PROPER(head(X)) → PROPER(X)
PROPER(tail(X)) → TAIL(proper(X))
PROPER(tail(X)) → PROPER(X)
PROPER(if(X1, X2, X3)) → IF(proper(X1), proper(X2), proper(X3))
PROPER(if(X1, X2, X3)) → PROPER(X1)
PROPER(if(X1, X2, X3)) → PROPER(X2)
PROPER(if(X1, X2, X3)) → PROPER(X3)
PROPER(filter(X1, X2)) → FILTER(proper(X1), proper(X2))
PROPER(filter(X1, X2)) → PROPER(X1)
PROPER(filter(X1, X2)) → PROPER(X2)
PROPER(divides(X1, X2)) → DIVIDES(proper(X1), proper(X2))
PROPER(divides(X1, X2)) → PROPER(X1)
PROPER(divides(X1, X2)) → PROPER(X2)
SIEVE(ok(X)) → SIEVE(X)
FROM(ok(X)) → FROM(X)
S(ok(X)) → S(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
HEAD(ok(X)) → HEAD(X)
TAIL(ok(X)) → TAIL(X)
IF(ok(X1), ok(X2), ok(X3)) → IF(X1, X2, X3)
FILTER(ok(X1), ok(X2)) → FILTER(X1, X2)
DIVIDES(ok(X1), ok(X2)) → DIVIDES(X1, X2)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 12 SCCs with 38 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DIVIDES(X1, mark(X2)) → DIVIDES(X1, X2)
DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
DIVIDES(ok(X1), ok(X2)) → DIVIDES(X1, X2)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

DIVIDES(X1, mark(X2)) → DIVIDES(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DIVIDES(x1, x2) = DIVIDES(x2)
mark(x1) = mark(x1)
ok(x1) = x1
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = x1
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

[active₁, filter₂] > [primes, 0] > mark₁ > top
[active₁, filter₂] > [cons₂, divides₂] > if₃ > mark₁ > top
true > mark₁ > top

Status:

DIVIDES₁: multiset
mark₁: [1]
active₁: [1]
primes: multiset
0: multiset
cons₂: multiset
if₃: [3,1,2]
true: multiset
false: multiset
filter₂: [1,2]
divides₂: [2,1]
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)
DIVIDES(ok(X1), ok(X2)) → DIVIDES(X1, X2)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

DIVIDES(mark(X1), X2) → DIVIDES(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DIVIDES(x1, x2) = x1
mark(x1) = mark(x1)
ok(x1) = x1
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = from(x1)
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > from₁ > cons₂ > divides₂ > mark₁
active₁ > 0
active₁ > filter₂ > cons₂ > divides₂ > mark₁
active₁ > filter₂ > if₃ > mark₁
primes > mark₁
primes > 0
true > mark₁
false > mark₁

Status:

mark₁: [1]
active₁: multiset
primes: multiset
from₁: [1]
0: multiset
cons₂: [1,2]
if₃: multiset
true: multiset
false: multiset
filter₂: [1,2]
divides₂: multiset
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DIVIDES(ok(X1), ok(X2)) → DIVIDES(X1, X2)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

DIVIDES(ok(X1), ok(X2)) → DIVIDES(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DIVIDES(x1, x2) = DIVIDES(x2)
ok(x1) = ok(x1)
active(x1) = x1
primes = primes
mark(x1) = mark
sieve(x1) = x1
from(x1) = x1
s(x1) = x1
0 = 0
cons(x1, x2) = x2
head(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = if(x1, x3)
true = true
false = false
filter(x1, x2) = x1
divides(x1, x2) = x1
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

DIVIDES₁ > mark
[true, proper₁] > primes > 0 > mark
[true, proper₁] > if₂ > ok₁ > top > mark
[true, proper₁] > false > mark

Status:

DIVIDES₁: multiset
ok₁: multiset
primes: multiset
mark: []
0: multiset
if₂: [2,1]
true: multiset
false: multiset
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(11) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(13) TRUE

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FILTER(X1, mark(X2)) → FILTER(X1, X2)
FILTER(mark(X1), X2) → FILTER(X1, X2)
FILTER(ok(X1), ok(X2)) → FILTER(X1, X2)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FILTER(X1, mark(X2)) → FILTER(X1, X2)
FILTER(mark(X1), X2) → FILTER(X1, X2)
FILTER(ok(X1), ok(X2)) → FILTER(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FILTER(x1, x2) = FILTER(x1, x2)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = x1
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

FILTER₂ > [mark₁, ok₁]
active₁ > [divides₂, proper₁] > cons₂ > [if₃, filter₂] > [mark₁, ok₁]
active₁ > [divides₂, proper₁] > true > [mark₁, ok₁]
primes > 0 > [mark₁, ok₁]
false > [mark₁, ok₁]
top > [mark₁, ok₁]

Status:

FILTER₂: [2,1]
mark₁: [1]
ok₁: [1]
active₁: multiset
primes: multiset
0: multiset
cons₂: [2,1]
if₃: [2,3,1]
true: multiset
false: multiset
filter₂: [1,2]
divides₂: [1,2]
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(16) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(18) TRUE

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IF(ok(X1), ok(X2), ok(X3)) → IF(X1, X2, X3)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(mark(X1), X2, X3) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3) = IF(x1)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
primes = primes
sieve(x1) = sieve(x1)
from(x1) = from(x1)
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = head(x1)
tail(x1) = x1
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > [sieve₁, cons₂] > filter₂ > if₃ > [mark₁, from₁]
active₁ > [sieve₁, cons₂] > filter₂ > divides₂ > [mark₁, from₁]
active₁ > 0
active₁ > head₁ > [mark₁, from₁]
true > [mark₁, from₁]

Status:

IF₁: multiset
mark₁: [1]
active₁: [1]
primes: multiset
sieve₁: multiset
from₁: [1]
0: multiset
cons₂: multiset
head₁: [1]
if₃: multiset
true: multiset
false: multiset
filter₂: multiset
divides₂: multiset
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IF(ok(X1), ok(X2), ok(X3)) → IF(X1, X2, X3)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(ok(X1), ok(X2), ok(X3)) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3) = x2
ok(x1) = ok(x1)
active(x1) = x1
primes = primes
mark(x1) = mark
sieve(x1) = sieve(x1)
from(x1) = x1
s(x1) = s(x1)
0 = 0
cons(x1, x2) = x2
head(x1) = head(x1)
tail(x1) = x1
if(x1, x2, x3) = if(x1)
true = true
false = false
filter(x1, x2) = filter(x1)
divides(x1, x2) = x2
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

0 > [ok₁, mark, filter₁]
true > [ok₁, mark, filter₁]
false > [ok₁, mark, filter₁]
top > [sieve₁, head₁, proper₁] > [primes, s₁] > [ok₁, mark, filter₁]
top > [sieve₁, head₁, proper₁] > if₁ > [ok₁, mark, filter₁]

Status:

ok₁: multiset
primes: multiset
mark: multiset
sieve₁: [1]
s₁: [1]
0: multiset
head₁: [1]
if₁: [1]
true: multiset
false: multiset
filter₁: multiset
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(23) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(24) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(25) TRUE

(26) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAIL(ok(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(ok(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAIL(x1) = x1
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = from(x1)
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = tail(x1)
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

proper₁ > [active₁, cons₂] > from₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > 0 > ok₁
proper₁ > [active₁, cons₂] > tail₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > if₃ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > divides₂ > [mark₁, true] > ok₁
proper₁ > primes > [mark₁, true] > ok₁
proper₁ > false > [mark₁, true] > ok₁

Status:

ok₁: [1]
mark₁: [1]
active₁: [1]
primes: multiset
from₁: [1]
0: multiset
cons₂: [1,2]
tail₁: [1]
if₃: multiset
true: multiset
false: multiset
filter₂: [1,2]
divides₂: multiset
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(28) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(29) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(30) TRUE

(31) Obligation:

Q DP problem:
The TRS P consists of the following rules:

HEAD(ok(X)) → HEAD(X)
HEAD(mark(X)) → HEAD(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(ok(X)) → HEAD(X)
HEAD(mark(X)) → HEAD(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
HEAD(x1) = x1
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = from(x1)
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = tail(x1)
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

proper₁ > [active₁, cons₂] > from₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > 0 > ok₁
proper₁ > [active₁, cons₂] > tail₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > if₃ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > divides₂ > [mark₁, true] > ok₁
proper₁ > primes > [mark₁, true] > ok₁
proper₁ > false > [mark₁, true] > ok₁

Status:

ok₁: [1]
mark₁: [1]
active₁: [1]
primes: multiset
from₁: [1]
0: multiset
cons₂: [1,2]
tail₁: [1]
if₃: multiset
true: multiset
false: multiset
filter₂: [1,2]
divides₂: multiset
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(33) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(34) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(35) TRUE

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(ok(X1), ok(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = x2
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = x1
primes = primes
sieve(x1) = x1
from(x1) = x1
s(x1) = s(x1)
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = if(x2, x3)
true = true
false = false
filter(x1, x2) = x2
divides(x1, x2) = x2
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

[primes, if₂, proper₁] > [s₁, cons₂] > [ok₁, top] > mark
[primes, if₂, proper₁] > 0 > [ok₁, top] > mark
[primes, if₂, proper₁] > true > [ok₁, top] > mark
[primes, if₂, proper₁] > false > mark

Status:

ok₁: [1]
mark: []
primes: multiset
s₁: [1]
0: multiset
cons₂: multiset
if₂: [2,1]
true: multiset
false: multiset
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1, x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = x1
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

[active₁, top] > [primes, 0]
[active₁, top] > filter₂ > cons₂ > [mark₁, false]
[active₁, top] > filter₂ > if₃ > [mark₁, false]
[active₁, top] > filter₂ > divides₂ > [mark₁, false]
true > [mark₁, false]

Status:

CONS₂: [1,2]
mark₁: [1]
active₁: [1]
primes: multiset
0: multiset
cons₂: [1,2]
if₃: [1,3,2]
true: multiset
false: multiset
filter₂: [2,1]
divides₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(40) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(41) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(42) TRUE

(43) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = x1
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = from(x1)
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = tail(x1)
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

proper₁ > [active₁, cons₂] > from₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > 0 > ok₁
proper₁ > [active₁, cons₂] > tail₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > if₃ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > divides₂ > [mark₁, true] > ok₁
proper₁ > primes > [mark₁, true] > ok₁
proper₁ > false > [mark₁, true] > ok₁

Status:

ok₁: [1]
mark₁: [1]
active₁: [1]
primes: multiset
from₁: [1]
0: multiset
cons₂: [1,2]
tail₁: [1]
if₃: multiset
true: multiset
false: multiset
filter₂: [1,2]
divides₂: multiset
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(45) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(46) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(47) TRUE

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FROM(ok(X)) → FROM(X)
FROM(mark(X)) → FROM(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(ok(X)) → FROM(X)
FROM(mark(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1) = x1
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = from(x1)
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = tail(x1)
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

proper₁ > [active₁, cons₂] > from₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > 0 > ok₁
proper₁ > [active₁, cons₂] > tail₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > if₃ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > divides₂ > [mark₁, true] > ok₁
proper₁ > primes > [mark₁, true] > ok₁
proper₁ > false > [mark₁, true] > ok₁

Status:

ok₁: [1]
mark₁: [1]
active₁: [1]
primes: multiset
from₁: [1]
0: multiset
cons₂: [1,2]
tail₁: [1]
if₃: multiset
true: multiset
false: multiset
filter₂: [1,2]
divides₂: multiset
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(50) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(51) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(52) TRUE

(53) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SIEVE(ok(X)) → SIEVE(X)
SIEVE(mark(X)) → SIEVE(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SIEVE(ok(X)) → SIEVE(X)
SIEVE(mark(X)) → SIEVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SIEVE(x1) = x1
ok(x1) = ok(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
primes = primes
sieve(x1) = x1
from(x1) = from(x1)
s(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
tail(x1) = tail(x1)
if(x1, x2, x3) = if(x1, x2, x3)
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

proper₁ > [active₁, cons₂] > from₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > 0 > ok₁
proper₁ > [active₁, cons₂] > tail₁ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > if₃ > [mark₁, true] > ok₁
proper₁ > [active₁, cons₂] > filter₂ > divides₂ > [mark₁, true] > ok₁
proper₁ > primes > [mark₁, true] > ok₁
proper₁ > false > [mark₁, true] > ok₁

Status:

ok₁: [1]
mark₁: [1]
active₁: [1]
primes: multiset
from₁: [1]
0: multiset
cons₂: [1,2]
tail₁: [1]
if₃: multiset
true: multiset
false: multiset
filter₂: [1,2]
divides₂: multiset
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(55) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(56) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(57) TRUE

(58) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(from(X)) → PROPER(X)
PROPER(sieve(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(head(X)) → PROPER(X)
PROPER(tail(X)) → PROPER(X)
PROPER(if(X1, X2, X3)) → PROPER(X1)
PROPER(if(X1, X2, X3)) → PROPER(X2)
PROPER(if(X1, X2, X3)) → PROPER(X3)
PROPER(filter(X1, X2)) → PROPER(X1)
PROPER(filter(X1, X2)) → PROPER(X2)
PROPER(divides(X1, X2)) → PROPER(X1)
PROPER(divides(X1, X2)) → PROPER(X2)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(sieve(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(head(X)) → PROPER(X)
PROPER(tail(X)) → PROPER(X)
PROPER(if(X1, X2, X3)) → PROPER(X1)
PROPER(if(X1, X2, X3)) → PROPER(X2)
PROPER(if(X1, X2, X3)) → PROPER(X3)
PROPER(filter(X1, X2)) → PROPER(X1)
PROPER(filter(X1, X2)) → PROPER(X2)
PROPER(divides(X1, X2)) → PROPER(X1)
PROPER(divides(X1, X2)) → PROPER(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
from(x1) = x1
sieve(x1) = sieve(x1)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
head(x1) = head(x1)
tail(x1) = tail(x1)
if(x1, x2, x3) = if(x1, x2, x3)
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
active(x1) = x1
primes = primes
mark(x1) = mark
0 = 0
true = true
false = false
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

[cons₂, filter₂] > [sieve₁, mark]
head₁ > [sieve₁, mark]
if₃ > [PROPER₁, tail₁] > [sieve₁, mark]
divides₂ > [PROPER₁, tail₁] > [sieve₁, mark]
primes > 0 > [sieve₁, mark]
true > [sieve₁, mark]
false > [sieve₁, mark]
top > [sieve₁, mark]

Status:

PROPER₁: [1]
sieve₁: [1]
cons₂: [2,1]
head₁: [1]
tail₁: [1]
if₃: [3,2,1]
filter₂: [2,1]
divides₂: [2,1]
primes: multiset
mark: []
0: multiset
true: multiset
false: multiset
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(60) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(from(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(from(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
from(x1) = from(x1)
s(x1) = s(x1)
active(x1) = active(x1)
primes = primes
mark(x1) = mark
sieve(x1) = x1
0 = 0
cons(x1, x2) = cons(x1)
head(x1) = x1
tail(x1) = tail
if(x1, x2, x3) = if(x1, x2)
true = true
false = false
filter(x1, x2) = filter
divides(x1, x2) = divides(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

[active₁, if₂] > [mark, tail, filter] > [PROPER₁, from₁] > s₁ > top
[active₁, if₂] > [mark, tail, filter] > cons₁ > s₁ > top
[active₁, if₂] > [mark, tail, filter] > cons₁ > divides₂ > top
[active₁, if₂] > 0 > top
primes > [PROPER₁, from₁] > s₁ > top
primes > 0 > top
true > top
false > top

Status:

PROPER₁: multiset
from₁: multiset
s₁: [1]
active₁: [1]
primes: multiset
mark: []
0: multiset
cons₁: [1]
tail: []
if₂: [2,1]
true: multiset
false: multiset
filter: []
divides₂: [1,2]
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(62) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(63) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(64) TRUE

(65) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(sieve(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)
ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(filter(X1, X2)) → ACTIVE(X1)
ACTIVE(filter(X1, X2)) → ACTIVE(X2)
ACTIVE(divides(X1, X2)) → ACTIVE(X1)
ACTIVE(divides(X1, X2)) → ACTIVE(X2)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(filter(X1, X2)) → ACTIVE(X1)
ACTIVE(filter(X1, X2)) → ACTIVE(X2)
ACTIVE(divides(X1, X2)) → ACTIVE(X1)
ACTIVE(divides(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
from(x1) = from(x1)
sieve(x1) = x1
s(x1) = x1
cons(x1, x2) = cons(x1)
head(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = x1
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
active(x1) = active(x1)
primes = primes
mark(x1) = mark
0 = 0
true = true
false = false
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > [cons₁, filter₂, proper₁] > ACTIVE₁
active₁ > [cons₁, filter₂, proper₁] > true
active₁ > [cons₁, filter₂, proper₁] > false > [primes, mark] > from₁
active₁ > [cons₁, filter₂, proper₁] > false > [primes, mark] > divides₂
active₁ > [cons₁, filter₂, proper₁] > false > [primes, mark] > top

Status:

ACTIVE₁: multiset
from₁: multiset
cons₁: [1]
filter₂: [1,2]
divides₂: [2,1]
active₁: [1]
primes: multiset
mark: []
0: multiset
true: multiset
false: multiset
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(67) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(sieve(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(head(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)
ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(head(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
sieve(x1) = x1
s(x1) = x1
head(x1) = head(x1)
tail(x1) = x1
if(x1, x2, x3) = x1
active(x1) = active(x1)
primes = primes
mark(x1) = mark
from(x1) = from
0 = 0
cons(x1, x2) = cons
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > [ACTIVE₁, head₁] > mark > top
active₁ > [from, cons, filter₂] > mark > top
active₁ > 0
active₁ > divides₁ > mark > top
false > mark > top

Status:

ACTIVE₁: [1]
head₁: multiset
active₁: [1]
primes: multiset
mark: []
from: []
0: multiset
cons: multiset
true: multiset
false: multiset
filter₂: [2,1]
divides₁: multiset
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(69) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(sieve(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)
ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
sieve(x1) = x1
s(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = if(x1, x2, x3)
active(x1) = active(x1)
primes = primes
mark(x1) = mark(x1)
from(x1) = x1
0 = 0
cons(x1, x2) = cons(x1, x2)
head(x1) = x1
true = true
false = false
filter(x1, x2) = filter(x1, x2)
divides(x1, x2) = divides(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

[active₁, divides₂] > [primes, 0] > mark₁ > top
[active₁, divides₂] > [cons₂, filter₂] > if₃ > mark₁ > top

Status:

ACTIVE₁: [1]
if₃: [2,3,1]
active₁: [1]
primes: multiset
mark₁: [1]
0: multiset
cons₂: [2,1]
true: multiset
false: multiset
filter₂: multiset
divides₂: [2,1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(71) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(sieve(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = x1
sieve(x1) = x1
s(x1) = s(x1)
tail(x1) = x1
active(x1) = active(x1)
primes = primes
mark(x1) = mark
from(x1) = from(x1)
0 = 0
cons(x1, x2) = x1
head(x1) = head
if(x1, x2, x3) = if(x1)
true = true
false = false
filter(x1, x2) = filter(x1)
divides(x1, x2) = divides(x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

head > [active₁, mark, filter₁, divides₁] > from₁ > [s₁, if₁]
head > [active₁, mark, filter₁, divides₁] > 0
top > [active₁, mark, filter₁, divides₁] > from₁ > [s₁, if₁]
top > [active₁, mark, filter₁, divides₁] > 0

Status:

s₁: multiset
active₁: [1]
primes: multiset
mark: []
from₁: [1]
0: multiset
head: []
if₁: [1]
true: multiset
false: multiset
filter₁: [1]
divides₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(73) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(sieve(X)) → ACTIVE(X)
ACTIVE(tail(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(tail(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = x1
sieve(x1) = x1
tail(x1) = tail(x1)
active(x1) = x1
primes = primes
mark(x1) = mark
from(x1) = x1
s(x1) = x1
0 = 0
cons(x1, x2) = cons
head(x1) = x1
if(x1, x2, x3) = x3
true = true
false = false
filter(x1, x2) = filter
divides(x1, x2) = divides
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

primes > [mark, false, filter]
0 > [mark, false, filter]
cons > [tail₁, proper₁] > true > [mark, false, filter]
divides > [tail₁, proper₁] > true > [mark, false, filter]
top > [tail₁, proper₁] > true > [mark, false, filter]

Status:

tail₁: multiset
primes: multiset
mark: multiset
0: multiset
cons: []
true: multiset
false: multiset
filter: multiset
divides: []
proper₁: multiset
top: []

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(75) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(sieve(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(sieve(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
sieve(x1) = sieve(x1)
active(x1) = x1
primes = primes
mark(x1) = mark
from(x1) = from(x1)
s(x1) = x1
0 = 0
cons(x1, x2) = cons
head(x1) = x1
tail(x1) = x1
if(x1, x2, x3) = if(x2)
true = true
false = false
filter(x1, x2) = filter
divides(x1, x2) = divides
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Recursive path order with status [RPO].
Quasi-Precedence:

ACTIVE₁ > sieve₁
[filter, proper₁] > 0 > [primes, ok] > [mark, from₁] > sieve₁
[filter, proper₁] > cons > divides > [primes, ok] > [mark, from₁] > sieve₁
[filter, proper₁] > if₁ > [primes, ok] > [mark, from₁] > sieve₁
[filter, proper₁] > true > [primes, ok] > [mark, from₁] > sieve₁
[filter, proper₁] > false > sieve₁
top > sieve₁

Status:

ACTIVE₁: multiset
sieve₁: multiset
primes: multiset
mark: []
from₁: [1]
0: multiset
cons: multiset
if₁: multiset
true: multiset
false: multiset
filter: []
divides: []
proper₁: [1]
ok: []
top: multiset

The following usable rules [FROCOS05] were oriented:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(77) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(78) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(79) TRUE

(80) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

The TRS R consists of the following rules:

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))))
active(sieve(X)) → sieve(active(X))
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(filter(X1, X2)) → filter(active(X1), X2)
active(filter(X1, X2)) → filter(X1, active(X2))
active(divides(X1, X2)) → divides(active(X1), X2)
active(divides(X1, X2)) → divides(X1, active(X2))
sieve(mark(X)) → mark(sieve(X))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
filter(mark(X1), X2) → mark(filter(X1, X2))
filter(X1, mark(X2)) → mark(filter(X1, X2))
divides(mark(X1), X2) → mark(divides(X1, X2))
divides(X1, mark(X2)) → mark(divides(X1, X2))
proper(primes) → ok(primes)
proper(sieve(X)) → sieve(proper(X))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(true) → ok(true)
proper(false) → ok(false)
proper(filter(X1, X2)) → filter(proper(X1), proper(X2))
proper(divides(X1, X2)) → divides(proper(X1), proper(X2))
sieve(ok(X)) → ok(sieve(X))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
filter(ok(X1), ok(X2)) → ok(filter(X1, X2))
divides(ok(X1), ok(X2)) → ok(divides(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.