Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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:

ACTIVE1(nats1(X)) -> NATS1(active1(X))
ACTIVE1(nats1(N)) -> S1(N)
S1(mark1(X)) -> S1(X)
PROPER1(nats1(X)) -> PROPER1(X)
ACTIVE1(zprimes) -> S1(0)
FILTER3(mark1(X1), X2, X3) -> FILTER3(X1, X2, X3)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X3)
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X2)
ACTIVE1(filter3(X1, X2, X3)) -> FILTER3(X1, X2, active1(X3))
TOP1(ok1(X)) -> ACTIVE1(X)
ACTIVE1(filter3(X1, X2, X3)) -> FILTER3(X1, active1(X2), X3)
ACTIVE1(sieve1(cons2(s1(N), Y))) -> CONS2(s1(N), sieve1(filter3(Y, N, N)))
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X3)
ACTIVE1(s1(X)) -> S1(active1(X))
ACTIVE1(sieve1(cons2(0, Y))) -> CONS2(0, sieve1(Y))
ACTIVE1(filter3(X1, X2, X3)) -> FILTER3(active1(X1), X2, X3)
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(sieve1(cons2(s1(N), Y))) -> SIEVE1(filter3(Y, N, N))
ACTIVE1(zprimes) -> SIEVE1(nats1(s1(s1(0))))
ACTIVE1(sieve1(cons2(0, Y))) -> SIEVE1(Y)
ACTIVE1(zprimes) -> S1(s1(0))
ACTIVE1(filter3(cons2(X, Y), 0, M)) -> CONS2(0, filter3(Y, M, M))
ACTIVE1(sieve1(X)) -> SIEVE1(active1(X))
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(zprimes) -> NATS1(s1(s1(0)))
FILTER3(X1, X2, mark1(X3)) -> FILTER3(X1, X2, X3)
FILTER3(ok1(X1), ok1(X2), ok1(X3)) -> FILTER3(X1, X2, X3)
FILTER3(X1, mark1(X2), X3) -> FILTER3(X1, X2, X3)
ACTIVE1(nats1(N)) -> CONS2(N, nats1(s1(N)))
PROPER1(sieve1(X)) -> SIEVE1(proper1(X))
NATS1(mark1(X)) -> NATS1(X)
PROPER1(s1(X)) -> S1(proper1(X))
PROPER1(sieve1(X)) -> PROPER1(X)
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
SIEVE1(ok1(X)) -> SIEVE1(X)
ACTIVE1(filter3(cons2(X, Y), s1(N), M)) -> FILTER3(Y, N, M)
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X1)
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X1)
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X2)
SIEVE1(mark1(X)) -> SIEVE1(X)
ACTIVE1(filter3(cons2(X, Y), 0, M)) -> FILTER3(Y, M, M)
ACTIVE1(nats1(X)) -> ACTIVE1(X)
ACTIVE1(sieve1(X)) -> ACTIVE1(X)
PROPER1(filter3(X1, X2, X3)) -> FILTER3(proper1(X1), proper1(X2), proper1(X3))
NATS1(ok1(X)) -> NATS1(X)
ACTIVE1(filter3(cons2(X, Y), s1(N), M)) -> CONS2(X, filter3(Y, N, M))
ACTIVE1(nats1(N)) -> NATS1(s1(N))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
ACTIVE1(sieve1(cons2(s1(N), Y))) -> FILTER3(Y, N, N)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(nats1(X)) -> NATS1(proper1(X))
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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:

ACTIVE1(nats1(X)) -> NATS1(active1(X))
ACTIVE1(nats1(N)) -> S1(N)
S1(mark1(X)) -> S1(X)
PROPER1(nats1(X)) -> PROPER1(X)
ACTIVE1(zprimes) -> S1(0)
FILTER3(mark1(X1), X2, X3) -> FILTER3(X1, X2, X3)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X3)
TOP1(mark1(X)) -> TOP1(proper1(X))
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X2)
ACTIVE1(filter3(X1, X2, X3)) -> FILTER3(X1, X2, active1(X3))
TOP1(ok1(X)) -> ACTIVE1(X)
ACTIVE1(filter3(X1, X2, X3)) -> FILTER3(X1, active1(X2), X3)
ACTIVE1(sieve1(cons2(s1(N), Y))) -> CONS2(s1(N), sieve1(filter3(Y, N, N)))
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X3)
ACTIVE1(s1(X)) -> S1(active1(X))
ACTIVE1(sieve1(cons2(0, Y))) -> CONS2(0, sieve1(Y))
ACTIVE1(filter3(X1, X2, X3)) -> FILTER3(active1(X1), X2, X3)
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(sieve1(cons2(s1(N), Y))) -> SIEVE1(filter3(Y, N, N))
ACTIVE1(zprimes) -> SIEVE1(nats1(s1(s1(0))))
ACTIVE1(sieve1(cons2(0, Y))) -> SIEVE1(Y)
ACTIVE1(zprimes) -> S1(s1(0))
ACTIVE1(filter3(cons2(X, Y), 0, M)) -> CONS2(0, filter3(Y, M, M))
ACTIVE1(sieve1(X)) -> SIEVE1(active1(X))
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(zprimes) -> NATS1(s1(s1(0)))
FILTER3(X1, X2, mark1(X3)) -> FILTER3(X1, X2, X3)
FILTER3(ok1(X1), ok1(X2), ok1(X3)) -> FILTER3(X1, X2, X3)
FILTER3(X1, mark1(X2), X3) -> FILTER3(X1, X2, X3)
ACTIVE1(nats1(N)) -> CONS2(N, nats1(s1(N)))
PROPER1(sieve1(X)) -> SIEVE1(proper1(X))
NATS1(mark1(X)) -> NATS1(X)
PROPER1(s1(X)) -> S1(proper1(X))
PROPER1(sieve1(X)) -> PROPER1(X)
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
SIEVE1(ok1(X)) -> SIEVE1(X)
ACTIVE1(filter3(cons2(X, Y), s1(N), M)) -> FILTER3(Y, N, M)
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X1)
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X1)
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X2)
SIEVE1(mark1(X)) -> SIEVE1(X)
ACTIVE1(filter3(cons2(X, Y), 0, M)) -> FILTER3(Y, M, M)
ACTIVE1(nats1(X)) -> ACTIVE1(X)
ACTIVE1(sieve1(X)) -> ACTIVE1(X)
PROPER1(filter3(X1, X2, X3)) -> FILTER3(proper1(X1), proper1(X2), proper1(X3))
NATS1(ok1(X)) -> NATS1(X)
ACTIVE1(filter3(cons2(X, Y), s1(N), M)) -> CONS2(X, filter3(Y, N, M))
ACTIVE1(nats1(N)) -> NATS1(s1(N))
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
ACTIVE1(sieve1(cons2(s1(N), Y))) -> FILTER3(Y, N, N)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(nats1(X)) -> NATS1(proper1(X))
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 8 SCCs with 30 less nodes.

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

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

NATS1(ok1(X)) -> NATS1(X)
NATS1(mark1(X)) -> NATS1(X)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


NATS1(mark1(X)) -> NATS1(X)
The remaining pairs can at least by weakly be oriented.

NATS1(ok1(X)) -> NATS1(X)
Used ordering: Combined order from the following AFS and order.
NATS1(x1)  =  NATS1(x1)
ok1(x1)  =  x1
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

NATS1(ok1(X)) -> NATS1(X)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


NATS1(ok1(X)) -> NATS1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
NATS1(x1)  =  NATS1(x1)
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
ok1 > NATS1


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ 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(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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

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(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


SIEVE1(ok1(X)) -> SIEVE1(X)
The remaining pairs can at least by weakly be oriented.

SIEVE1(mark1(X)) -> SIEVE1(X)
Used ordering: Combined order from the following AFS and order.
SIEVE1(x1)  =  SIEVE1(x1)
mark1(x1)  =  x1
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

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

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


SIEVE1(mark1(X)) -> SIEVE1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
SIEVE1(x1)  =  SIEVE1(x1)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
mark1 > SIEVE1


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ 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(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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

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(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


S1(mark1(X)) -> S1(X)
The remaining pairs can at least by weakly be oriented.

S1(ok1(X)) -> S1(X)
Used ordering: Combined order from the following AFS and order.
S1(x1)  =  S1(x1)
ok1(x1)  =  x1
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

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

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


S1(ok1(X)) -> S1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
S1(x1)  =  S1(x1)
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
ok1 > S1


The following usable rules [14] were oriented: none



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

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

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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

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(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least by weakly be oriented.

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
Used ordering: Combined order from the following AFS and order.
CONS2(x1, x2)  =  x2
mark1(x1)  =  mark
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

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

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


CONS2(mark1(X1), X2) -> CONS2(X1, X2)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
CONS2(x1, x2)  =  CONS1(x1)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
[CONS1, mark1]


The following usable rules [14] were oriented: none



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

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

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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

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

FILTER3(mark1(X1), X2, X3) -> FILTER3(X1, X2, X3)
FILTER3(X1, X2, mark1(X3)) -> FILTER3(X1, X2, X3)
FILTER3(ok1(X1), ok1(X2), ok1(X3)) -> FILTER3(X1, X2, X3)
FILTER3(X1, mark1(X2), X3) -> FILTER3(X1, X2, X3)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


FILTER3(ok1(X1), ok1(X2), ok1(X3)) -> FILTER3(X1, X2, X3)
The remaining pairs can at least by weakly be oriented.

FILTER3(mark1(X1), X2, X3) -> FILTER3(X1, X2, X3)
FILTER3(X1, X2, mark1(X3)) -> FILTER3(X1, X2, X3)
FILTER3(X1, mark1(X2), X3) -> FILTER3(X1, X2, X3)
Used ordering: Combined order from the following AFS and order.
FILTER3(x1, x2, x3)  =  FILTER1(x3)
mark1(x1)  =  x1
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

FILTER3(mark1(X1), X2, X3) -> FILTER3(X1, X2, X3)
FILTER3(X1, X2, mark1(X3)) -> FILTER3(X1, X2, X3)
FILTER3(X1, mark1(X2), X3) -> FILTER3(X1, X2, X3)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


FILTER3(mark1(X1), X2, X3) -> FILTER3(X1, X2, X3)
FILTER3(X1, X2, mark1(X3)) -> FILTER3(X1, X2, X3)
The remaining pairs can at least by weakly be oriented.

FILTER3(X1, mark1(X2), X3) -> FILTER3(X1, X2, X3)
Used ordering: Combined order from the following AFS and order.
FILTER3(x1, x2, x3)  =  FILTER2(x1, x3)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
[FILTER2, mark1]


The following usable rules [14] were oriented: none



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

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

FILTER3(X1, mark1(X2), X3) -> FILTER3(X1, X2, X3)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


FILTER3(X1, mark1(X2), X3) -> FILTER3(X1, X2, X3)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
FILTER3(x1, x2, x3)  =  FILTER1(x2)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
[FILTER1, mark1]


The following usable rules [14] were oriented: none



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

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

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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

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

PROPER1(filter3(X1, X2, X3)) -> PROPER1(X3)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X1)
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X2)
PROPER1(sieve1(X)) -> PROPER1(X)
PROPER1(nats1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


PROPER1(filter3(X1, X2, X3)) -> PROPER1(X3)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X1)
PROPER1(filter3(X1, X2, X3)) -> PROPER1(X2)
PROPER1(sieve1(X)) -> PROPER1(X)
The remaining pairs can at least by weakly be oriented.

PROPER1(s1(X)) -> PROPER1(X)
PROPER1(nats1(X)) -> PROPER1(X)
Used ordering: Combined order from the following AFS and order.
PROPER1(x1)  =  PROPER1(x1)
filter3(x1, x2, x3)  =  filter3(x1, x2, x3)
s1(x1)  =  x1
cons2(x1, x2)  =  cons2(x1, x2)
sieve1(x1)  =  sieve1(x1)
nats1(x1)  =  x1

Lexicographic Path Order [19].
Precedence:
cons2 > PROPER1


The following usable rules [14] were oriented: none



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

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

PROPER1(s1(X)) -> PROPER1(X)
PROPER1(nats1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


PROPER1(nats1(X)) -> PROPER1(X)
The remaining pairs can at least by weakly be oriented.

PROPER1(s1(X)) -> PROPER1(X)
Used ordering: Combined order from the following AFS and order.
PROPER1(x1)  =  PROPER1(x1)
s1(x1)  =  x1
nats1(x1)  =  nats1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

PROPER1(s1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


PROPER1(s1(X)) -> PROPER1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
PROPER1(x1)  =  PROPER1(x1)
s1(x1)  =  s1(x1)

Lexicographic Path Order [19].
Precedence:
s1 > PROPER1


The following usable rules [14] were oriented: none



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

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

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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

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

ACTIVE1(sieve1(X)) -> ACTIVE1(X)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X3)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X1)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(nats1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


ACTIVE1(sieve1(X)) -> ACTIVE1(X)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X3)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X1)
ACTIVE1(filter3(X1, X2, X3)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
The remaining pairs can at least by weakly be oriented.

ACTIVE1(nats1(X)) -> ACTIVE1(X)
Used ordering: Combined order from the following AFS and order.
ACTIVE1(x1)  =  ACTIVE1(x1)
sieve1(x1)  =  sieve1(x1)
filter3(x1, x2, x3)  =  filter3(x1, x2, x3)
cons2(x1, x2)  =  cons2(x1, x2)
s1(x1)  =  s1(x1)
nats1(x1)  =  x1

Lexicographic Path Order [19].
Precedence:
sieve1 > [ACTIVE1, cons2]
filter3 > [ACTIVE1, cons2]
s1 > [ACTIVE1, cons2]


The following usable rules [14] were oriented: none



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

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

ACTIVE1(nats1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


ACTIVE1(nats1(X)) -> ACTIVE1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
ACTIVE1(x1)  =  ACTIVE1(x1)
nats1(x1)  =  nats1(x1)

Lexicographic Path Order [19].
Precedence:
nats1 > ACTIVE1


The following usable rules [14] were oriented: none



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

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

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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

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(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


TOP1(mark1(X)) -> TOP1(proper1(X))
The remaining pairs can at least by weakly be oriented.

TOP1(ok1(X)) -> TOP1(active1(X))
Used ordering: Combined order from the following AFS and order.
TOP1(x1)  =  TOP1(x1)
ok1(x1)  =  x1
active1(x1)  =  x1
mark1(x1)  =  mark1(x1)
proper1(x1)  =  x1
filter3(x1, x2, x3)  =  filter3(x1, x2, x3)
cons2(x1, x2)  =  x1
0  =  0
s1(x1)  =  s1(x1)
sieve1(x1)  =  sieve1(x1)
nats1(x1)  =  nats1(x1)
zprimes  =  zprimes

Lexicographic Path Order [19].
Precedence:
zprimes > [filter3, sieve1] > 0 > [mark1, s1]
zprimes > nats1 > [mark1, s1]


The following usable rules [14] were oriented:

proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
nats1(mark1(X)) -> mark1(nats1(X))
nats1(ok1(X)) -> ok1(nats1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
s1(mark1(X)) -> mark1(s1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))



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

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

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

The TRS R consists of the following rules:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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 strictly oriented and are deleted.


TOP1(ok1(X)) -> TOP1(active1(X))
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
TOP1(x1)  =  TOP1(x1)
ok1(x1)  =  ok1(x1)
active1(x1)  =  active1(x1)
filter3(x1, x2, x3)  =  x1
cons2(x1, x2)  =  x1
0  =  0
mark1(x1)  =  mark
s1(x1)  =  x1
sieve1(x1)  =  x1
nats1(x1)  =  x1
zprimes  =  zprimes

Lexicographic Path Order [19].
Precedence:
[TOP1, ok1] > active1 > 0 > [mark, zprimes]


The following usable rules [14] were oriented:

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
nats1(mark1(X)) -> mark1(nats1(X))
nats1(ok1(X)) -> ok1(nats1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
s1(mark1(X)) -> mark1(s1(X))
s1(ok1(X)) -> ok1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))



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

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

active1(filter3(cons2(X, Y), 0, M)) -> mark1(cons2(0, filter3(Y, M, M)))
active1(filter3(cons2(X, Y), s1(N), M)) -> mark1(cons2(X, filter3(Y, N, M)))
active1(sieve1(cons2(0, Y))) -> mark1(cons2(0, sieve1(Y)))
active1(sieve1(cons2(s1(N), Y))) -> mark1(cons2(s1(N), sieve1(filter3(Y, N, N))))
active1(nats1(N)) -> mark1(cons2(N, nats1(s1(N))))
active1(zprimes) -> mark1(sieve1(nats1(s1(s1(0)))))
active1(filter3(X1, X2, X3)) -> filter3(active1(X1), X2, X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, active1(X2), X3)
active1(filter3(X1, X2, X3)) -> filter3(X1, X2, active1(X3))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sieve1(X)) -> sieve1(active1(X))
active1(nats1(X)) -> nats1(active1(X))
filter3(mark1(X1), X2, X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, mark1(X2), X3) -> mark1(filter3(X1, X2, X3))
filter3(X1, X2, mark1(X3)) -> mark1(filter3(X1, X2, X3))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sieve1(mark1(X)) -> mark1(sieve1(X))
nats1(mark1(X)) -> mark1(nats1(X))
proper1(filter3(X1, X2, X3)) -> filter3(proper1(X1), proper1(X2), proper1(X3))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(sieve1(X)) -> sieve1(proper1(X))
proper1(nats1(X)) -> nats1(proper1(X))
proper1(zprimes) -> ok1(zprimes)
filter3(ok1(X1), ok1(X2), ok1(X3)) -> ok1(filter3(X1, X2, X3))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sieve1(ok1(X)) -> ok1(sieve1(X))
nats1(ok1(X)) -> ok1(nats1(X))
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.