Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(from1(X)) -> FROM1(s1(X))
FILTER2(mark1(X1), X2) -> FILTER2(X1, X2)
ACTIVE1(divides2(X1, X2)) -> ACTIVE1(X2)
PROPER1(filter2(X1, X2)) -> PROPER1(X2)
ACTIVE1(divides2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> FROM1(active1(X))
PROPER1(divides2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
S1(mark1(X)) -> S1(X)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> DIVIDES2(s1(s1(X)), Y)
ACTIVE1(if3(X1, X2, X3)) -> IF3(active1(X1), X2, X3)
PROPER1(divides2(X1, X2)) -> DIVIDES2(proper1(X1), proper1(X2))
ACTIVE1(sieve1(cons2(X, Y))) -> CONS2(X, filter2(X, sieve1(Y)))
TOP1(mark1(X)) -> TOP1(proper1(X))
ACTIVE1(tail1(X)) -> TAIL1(active1(X))
IF3(mark1(X1), X2, X3) -> IF3(X1, X2, X3)
HEAD1(mark1(X)) -> HEAD1(X)
ACTIVE1(filter2(X1, X2)) -> FILTER2(active1(X1), X2)
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(divides2(X1, X2)) -> DIVIDES2(X1, active1(X2))
PROPER1(tail1(X)) -> PROPER1(X)
TOP1(ok1(X)) -> ACTIVE1(X)
DIVIDES2(ok1(X1), ok1(X2)) -> DIVIDES2(X1, X2)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(divides2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(tail1(X)) -> TAIL1(proper1(X))
ACTIVE1(primes) -> S1(s1(0))
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> SIEVE1(Y)
ACTIVE1(divides2(X1, X2)) -> DIVIDES2(active1(X1), X2)
DIVIDES2(mark1(X1), X2) -> DIVIDES2(X1, X2)
ACTIVE1(primes) -> S1(0)
HEAD1(ok1(X)) -> HEAD1(X)
ACTIVE1(tail1(X)) -> ACTIVE1(X)
ACTIVE1(sieve1(X)) -> SIEVE1(active1(X))
TOP1(mark1(X)) -> PROPER1(X)
FILTER2(X1, mark1(X2)) -> FILTER2(X1, X2)
TAIL1(ok1(X)) -> TAIL1(X)
PROPER1(filter2(X1, X2)) -> PROPER1(X1)
PROPER1(if3(X1, X2, X3)) -> PROPER1(X1)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> CONS2(Y, filter2(X, sieve1(Y)))
ACTIVE1(primes) -> SIEVE1(from1(s1(s1(0))))
DIVIDES2(X1, mark1(X2)) -> DIVIDES2(X1, X2)
IF3(ok1(X1), ok1(X2), ok1(X3)) -> IF3(X1, X2, X3)
ACTIVE1(sieve1(cons2(X, Y))) -> SIEVE1(Y)
ACTIVE1(head1(X)) -> ACTIVE1(X)
PROPER1(if3(X1, X2, X3)) -> IF3(proper1(X1), proper1(X2), proper1(X3))
PROPER1(sieve1(X)) -> SIEVE1(proper1(X))
ACTIVE1(filter2(X1, X2)) -> ACTIVE1(X1)
PROPER1(filter2(X1, X2)) -> FILTER2(proper1(X1), proper1(X2))
PROPER1(if3(X1, X2, X3)) -> PROPER1(X3)
PROPER1(s1(X)) -> S1(proper1(X))
ACTIVE1(filter2(X1, X2)) -> ACTIVE1(X2)
PROPER1(sieve1(X)) -> PROPER1(X)
S1(ok1(X)) -> S1(X)
ACTIVE1(head1(X)) -> HEAD1(active1(X))
ACTIVE1(s1(X)) -> ACTIVE1(X)
SIEVE1(ok1(X)) -> SIEVE1(X)
ACTIVE1(filter2(X1, X2)) -> FILTER2(X1, active1(X2))
PROPER1(if3(X1, X2, X3)) -> PROPER1(X2)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> IF3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y))))
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
TAIL1(mark1(X)) -> TAIL1(X)
PROPER1(head1(X)) -> HEAD1(proper1(X))
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
SIEVE1(mark1(X)) -> SIEVE1(X)
FILTER2(ok1(X1), ok1(X2)) -> FILTER2(X1, X2)
ACTIVE1(sieve1(X)) -> ACTIVE1(X)
ACTIVE1(sieve1(cons2(X, Y))) -> FILTER2(X, sieve1(Y))
ACTIVE1(from1(X)) -> S1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(from1(X)) -> PROPER1(X)
ACTIVE1(if3(X1, X2, X3)) -> ACTIVE1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> FILTER2(s1(s1(X)), Z)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> FILTER2(X, sieve1(Y))
FROM1(ok1(X)) -> FROM1(X)
ACTIVE1(primes) -> FROM1(s1(s1(0)))

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(from1(X)) -> FROM1(s1(X))
FILTER2(mark1(X1), X2) -> FILTER2(X1, X2)
ACTIVE1(divides2(X1, X2)) -> ACTIVE1(X2)
PROPER1(filter2(X1, X2)) -> PROPER1(X2)
ACTIVE1(divides2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> FROM1(active1(X))
PROPER1(divides2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
S1(mark1(X)) -> S1(X)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> DIVIDES2(s1(s1(X)), Y)
ACTIVE1(if3(X1, X2, X3)) -> IF3(active1(X1), X2, X3)
PROPER1(divides2(X1, X2)) -> DIVIDES2(proper1(X1), proper1(X2))
ACTIVE1(sieve1(cons2(X, Y))) -> CONS2(X, filter2(X, sieve1(Y)))
TOP1(mark1(X)) -> TOP1(proper1(X))
ACTIVE1(tail1(X)) -> TAIL1(active1(X))
IF3(mark1(X1), X2, X3) -> IF3(X1, X2, X3)
HEAD1(mark1(X)) -> HEAD1(X)
ACTIVE1(filter2(X1, X2)) -> FILTER2(active1(X1), X2)
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(divides2(X1, X2)) -> DIVIDES2(X1, active1(X2))
PROPER1(tail1(X)) -> PROPER1(X)
TOP1(ok1(X)) -> ACTIVE1(X)
DIVIDES2(ok1(X1), ok1(X2)) -> DIVIDES2(X1, X2)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(divides2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(tail1(X)) -> TAIL1(proper1(X))
ACTIVE1(primes) -> S1(s1(0))
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> SIEVE1(Y)
ACTIVE1(divides2(X1, X2)) -> DIVIDES2(active1(X1), X2)
DIVIDES2(mark1(X1), X2) -> DIVIDES2(X1, X2)
ACTIVE1(primes) -> S1(0)
HEAD1(ok1(X)) -> HEAD1(X)
ACTIVE1(tail1(X)) -> ACTIVE1(X)
ACTIVE1(sieve1(X)) -> SIEVE1(active1(X))
TOP1(mark1(X)) -> PROPER1(X)
FILTER2(X1, mark1(X2)) -> FILTER2(X1, X2)
TAIL1(ok1(X)) -> TAIL1(X)
PROPER1(filter2(X1, X2)) -> PROPER1(X1)
PROPER1(if3(X1, X2, X3)) -> PROPER1(X1)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> CONS2(Y, filter2(X, sieve1(Y)))
ACTIVE1(primes) -> SIEVE1(from1(s1(s1(0))))
DIVIDES2(X1, mark1(X2)) -> DIVIDES2(X1, X2)
IF3(ok1(X1), ok1(X2), ok1(X3)) -> IF3(X1, X2, X3)
ACTIVE1(sieve1(cons2(X, Y))) -> SIEVE1(Y)
ACTIVE1(head1(X)) -> ACTIVE1(X)
PROPER1(if3(X1, X2, X3)) -> IF3(proper1(X1), proper1(X2), proper1(X3))
PROPER1(sieve1(X)) -> SIEVE1(proper1(X))
ACTIVE1(filter2(X1, X2)) -> ACTIVE1(X1)
PROPER1(filter2(X1, X2)) -> FILTER2(proper1(X1), proper1(X2))
PROPER1(if3(X1, X2, X3)) -> PROPER1(X3)
PROPER1(s1(X)) -> S1(proper1(X))
ACTIVE1(filter2(X1, X2)) -> ACTIVE1(X2)
PROPER1(sieve1(X)) -> PROPER1(X)
S1(ok1(X)) -> S1(X)
ACTIVE1(head1(X)) -> HEAD1(active1(X))
ACTIVE1(s1(X)) -> ACTIVE1(X)
SIEVE1(ok1(X)) -> SIEVE1(X)
ACTIVE1(filter2(X1, X2)) -> FILTER2(X1, active1(X2))
PROPER1(if3(X1, X2, X3)) -> PROPER1(X2)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> IF3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y))))
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
TAIL1(mark1(X)) -> TAIL1(X)
PROPER1(head1(X)) -> HEAD1(proper1(X))
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
SIEVE1(mark1(X)) -> SIEVE1(X)
FILTER2(ok1(X1), ok1(X2)) -> FILTER2(X1, X2)
ACTIVE1(sieve1(X)) -> ACTIVE1(X)
ACTIVE1(sieve1(cons2(X, Y))) -> FILTER2(X, sieve1(Y))
ACTIVE1(from1(X)) -> S1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(from1(X)) -> PROPER1(X)
ACTIVE1(if3(X1, X2, X3)) -> ACTIVE1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> FILTER2(s1(s1(X)), Z)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(filter2(s1(s1(X)), cons2(Y, Z))) -> FILTER2(X, sieve1(Y))
FROM1(ok1(X)) -> FROM1(X)
ACTIVE1(primes) -> FROM1(s1(s1(0)))

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 12 SCCs with 38 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

DIVIDES2(mark1(X1), X2) -> DIVIDES2(X1, X2)
DIVIDES2(X1, mark1(X2)) -> DIVIDES2(X1, X2)
DIVIDES2(ok1(X1), ok1(X2)) -> DIVIDES2(X1, X2)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


DIVIDES2(mark1(X1), X2) -> DIVIDES2(X1, X2)
DIVIDES2(X1, mark1(X2)) -> DIVIDES2(X1, X2)
DIVIDES2(ok1(X1), ok1(X2)) -> DIVIDES2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(DIVIDES2(x1, x2)) = 2·x1 + x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FILTER2(ok1(X1), ok1(X2)) -> FILTER2(X1, X2)
FILTER2(mark1(X1), X2) -> FILTER2(X1, X2)
FILTER2(X1, mark1(X2)) -> FILTER2(X1, X2)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FILTER2(ok1(X1), ok1(X2)) -> FILTER2(X1, X2)
FILTER2(mark1(X1), X2) -> FILTER2(X1, X2)
FILTER2(X1, mark1(X2)) -> FILTER2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FILTER2(x1, x2)) = 2·x1 + x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

IF3(mark1(X1), X2, X3) -> IF3(X1, X2, X3)
IF3(ok1(X1), ok1(X2), ok1(X3)) -> IF3(X1, X2, X3)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


IF3(mark1(X1), X2, X3) -> IF3(X1, X2, X3)
IF3(ok1(X1), ok1(X2), ok1(X3)) -> IF3(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(IF3(x1, x2, x3)) = 2·x1 + x2 + x3   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

TAIL1(mark1(X)) -> TAIL1(X)
TAIL1(ok1(X)) -> TAIL1(X)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TAIL1(mark1(X)) -> TAIL1(X)
TAIL1(ok1(X)) -> TAIL1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(TAIL1(x1)) = 2·x1   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

HEAD1(ok1(X)) -> HEAD1(X)
HEAD1(mark1(X)) -> HEAD1(X)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


HEAD1(ok1(X)) -> HEAD1(X)
HEAD1(mark1(X)) -> HEAD1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(HEAD1(x1)) = 2·x1   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = 2·x1 + x2   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(S1(x1)) = 2·x1   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FROM1(mark1(X)) -> FROM1(X)
FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FROM1(mark1(X)) -> FROM1(X)
FROM1(ok1(X)) -> FROM1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FROM1(x1)) = 2·x1   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SIEVE1(mark1(X)) -> SIEVE1(X)
SIEVE1(ok1(X)) -> SIEVE1(X)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SIEVE1(mark1(X)) -> SIEVE1(X)
SIEVE1(ok1(X)) -> SIEVE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SIEVE1(x1)) = 2·x1   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(filter2(X1, X2)) -> PROPER1(X2)
PROPER1(filter2(X1, X2)) -> PROPER1(X1)
PROPER1(tail1(X)) -> PROPER1(X)
PROPER1(if3(X1, X2, X3)) -> PROPER1(X1)
PROPER1(divides2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(divides2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(if3(X1, X2, X3)) -> PROPER1(X3)
PROPER1(sieve1(X)) -> PROPER1(X)
PROPER1(if3(X1, X2, X3)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(filter2(X1, X2)) -> PROPER1(X2)
PROPER1(filter2(X1, X2)) -> PROPER1(X1)
PROPER1(tail1(X)) -> PROPER1(X)
PROPER1(if3(X1, X2, X3)) -> PROPER1(X1)
PROPER1(divides2(X1, X2)) -> PROPER1(X2)
PROPER1(head1(X)) -> PROPER1(X)
PROPER1(divides2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(if3(X1, X2, X3)) -> PROPER1(X3)
PROPER1(sieve1(X)) -> PROPER1(X)
PROPER1(if3(X1, X2, X3)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = 2·x1   
POL(cons2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(divides2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(filter2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(from1(x1)) = 2 + 2·x1   
POL(head1(x1)) = 2 + 2·x1   
POL(if3(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3   
POL(s1(x1)) = 2 + 2·x1   
POL(sieve1(x1)) = 2 + 2·x1   
POL(tail1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(sieve1(X)) -> ACTIVE1(X)
ACTIVE1(head1(X)) -> ACTIVE1(X)
ACTIVE1(filter2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(divides2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(if3(X1, X2, X3)) -> ACTIVE1(X1)
ACTIVE1(divides2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(filter2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(tail1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(sieve1(X)) -> ACTIVE1(X)
ACTIVE1(head1(X)) -> ACTIVE1(X)
ACTIVE1(filter2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(divides2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(if3(X1, X2, X3)) -> ACTIVE1(X1)
ACTIVE1(divides2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(filter2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(tail1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = 2·x1   
POL(cons2(x1, x2)) = 2 + 2·x1   
POL(divides2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(filter2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(from1(x1)) = 2 + 2·x1   
POL(head1(x1)) = 2 + 2·x1   
POL(if3(x1, x2, x3)) = 2 + 2·x1   
POL(s1(x1)) = 2 + 2·x1   
POL(sieve1(x1)) = 2 + 2·x1   
POL(tail1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP

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

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

TOP1(ok1(X)) -> TOP1(active1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))

The TRS R consists of the following rules:

active1(primes) -> mark1(sieve1(from1(s1(s1(0)))))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(head1(cons2(X, Y))) -> mark1(X)
active1(tail1(cons2(X, Y))) -> mark1(Y)
active1(if3(true, X, Y)) -> mark1(X)
active1(if3(false, X, Y)) -> mark1(Y)
active1(filter2(s1(s1(X)), cons2(Y, Z))) -> mark1(if3(divides2(s1(s1(X)), Y), filter2(s1(s1(X)), Z), cons2(Y, filter2(X, sieve1(Y)))))
active1(sieve1(cons2(X, Y))) -> mark1(cons2(X, filter2(X, sieve1(Y))))
active1(sieve1(X)) -> sieve1(active1(X))
active1(from1(X)) -> from1(active1(X))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(head1(X)) -> head1(active1(X))
active1(tail1(X)) -> tail1(active1(X))
active1(if3(X1, X2, X3)) -> if3(active1(X1), X2, X3)
active1(filter2(X1, X2)) -> filter2(active1(X1), X2)
active1(filter2(X1, X2)) -> filter2(X1, active1(X2))
active1(divides2(X1, X2)) -> divides2(active1(X1), X2)
active1(divides2(X1, X2)) -> divides2(X1, active1(X2))
sieve1(mark1(X)) -> mark1(sieve1(X))
from1(mark1(X)) -> mark1(from1(X))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
head1(mark1(X)) -> mark1(head1(X))
tail1(mark1(X)) -> mark1(tail1(X))
if3(mark1(X1), X2, X3) -> mark1(if3(X1, X2, X3))
filter2(mark1(X1), X2) -> mark1(filter2(X1, X2))
filter2(X1, mark1(X2)) -> mark1(filter2(X1, X2))
divides2(mark1(X1), X2) -> mark1(divides2(X1, X2))
divides2(X1, mark1(X2)) -> mark1(divides2(X1, X2))
proper1(primes) -> ok1(primes)
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(from1(X)) -> from1(proper1(X))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(head1(X)) -> head1(proper1(X))
proper1(tail1(X)) -> tail1(proper1(X))
proper1(if3(X1, X2, X3)) -> if3(proper1(X1), proper1(X2), proper1(X3))
proper1(true) -> ok1(true)
proper1(false) -> ok1(false)
proper1(filter2(X1, X2)) -> filter2(proper1(X1), proper1(X2))
proper1(divides2(X1, X2)) -> divides2(proper1(X1), proper1(X2))
sieve1(ok1(X)) -> ok1(sieve1(X))
from1(ok1(X)) -> ok1(from1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
head1(ok1(X)) -> ok1(head1(X))
tail1(ok1(X)) -> ok1(tail1(X))
if3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(if3(X1, X2, X3))
filter2(ok1(X1), ok1(X2)) -> ok1(filter2(X1, X2))
divides2(ok1(X1), ok1(X2)) -> ok1(divides2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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