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:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

Q is empty.


QTRS
  ↳ Non-Overlap Check

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

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS
  ↳ Non-Overlap Check
QTRS
      ↳ DependencyPairsProof

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

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes


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

SIEVE1(cons2(s1(N), Y)) -> SIEVE1(filter3(Y, N, N))
ZPRIMES -> SIEVE1(nats1(s1(s1(0))))
SIEVE1(cons2(s1(N), Y)) -> FILTER3(Y, N, N)
FILTER3(cons2(X, Y), s1(N), M) -> FILTER3(Y, N, M)
FILTER3(cons2(X, Y), 0, M) -> FILTER3(Y, M, M)
SIEVE1(cons2(0, Y)) -> SIEVE1(Y)
ZPRIMES -> NATS1(s1(s1(0)))
NATS1(N) -> NATS1(s1(N))

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ DependencyGraphProof

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

SIEVE1(cons2(s1(N), Y)) -> SIEVE1(filter3(Y, N, N))
ZPRIMES -> SIEVE1(nats1(s1(s1(0))))
SIEVE1(cons2(s1(N), Y)) -> FILTER3(Y, N, N)
FILTER3(cons2(X, Y), s1(N), M) -> FILTER3(Y, N, M)
FILTER3(cons2(X, Y), 0, M) -> FILTER3(Y, M, M)
SIEVE1(cons2(0, Y)) -> SIEVE1(Y)
ZPRIMES -> NATS1(s1(s1(0)))
NATS1(N) -> NATS1(s1(N))

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 3 less nodes.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
QDP
              ↳ QDP
              ↳ QDP

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

NATS1(N) -> NATS1(s1(N))

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
QDP
                ↳ QDPOrderProof
              ↳ QDP

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

FILTER3(cons2(X, Y), 0, M) -> FILTER3(Y, M, M)
FILTER3(cons2(X, Y), s1(N), M) -> FILTER3(Y, N, M)

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes

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(cons2(X, Y), 0, M) -> FILTER3(Y, M, M)
FILTER3(cons2(X, Y), s1(N), M) -> FILTER3(Y, N, M)
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)  =  x1
cons2(x1, x2)  =  cons2(x1, x2)
0  =  0
s1(x1)  =  s

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
              ↳ QDP

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

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
QDP
                ↳ QDPOrderProof

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

SIEVE1(cons2(s1(N), Y)) -> SIEVE1(filter3(Y, N, N))
SIEVE1(cons2(0, Y)) -> SIEVE1(Y)

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes

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(cons2(0, Y)) -> SIEVE1(Y)
The remaining pairs can at least by weakly be oriented.

SIEVE1(cons2(s1(N), Y)) -> SIEVE1(filter3(Y, N, N))
Used ordering: Combined order from the following AFS and order.
SIEVE1(x1)  =  SIEVE1(x1)
cons2(x1, x2)  =  cons1(x2)
s1(x1)  =  s1(x1)
filter3(x1, x2, x3)  =  filter1(x1)
0  =  0

Lexicographic Path Order [19].
Precedence:
[SIEVE1, cons1, filter1] > 0


The following usable rules [14] were oriented:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof

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

SIEVE1(cons2(s1(N), Y)) -> SIEVE1(filter3(Y, N, N))

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes

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(cons2(s1(N), Y)) -> SIEVE1(filter3(Y, N, N))
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
SIEVE1(x1)  =  x1
cons2(x1, x2)  =  cons1(x2)
s1(x1)  =  s
filter3(x1, x2, x3)  =  x1
0  =  0

Lexicographic Path Order [19].
Precedence:
s > cons1
0 > cons1


The following usable rules [14] were oriented:

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof

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

filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, filter3(Y, N, M))
sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), sieve1(filter3(Y, N, N)))
nats1(N) -> cons2(N, nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))

The set Q consists of the following terms:

filter3(cons2(x0, x1), 0, x2)
filter3(cons2(x0, x1), s1(x2), x3)
sieve1(cons2(0, x0))
sieve1(cons2(s1(x0), x1))
nats1(x0)
zprimes

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