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:

filter3(cons2(X, Y), 0, M) -> cons2(0, n__filter3(activate1(Y), M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, n__filter3(activate1(Y), N, M))
sieve1(cons2(0, Y)) -> cons2(0, n__sieve1(activate1(Y)))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), n__sieve1(filter3(activate1(Y), N, N)))
nats1(N) -> cons2(N, n__nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))
filter3(X1, X2, X3) -> n__filter3(X1, X2, X3)
sieve1(X) -> n__sieve1(X)
nats1(X) -> n__nats1(X)
activate1(n__filter3(X1, X2, X3)) -> filter3(X1, X2, X3)
activate1(n__sieve1(X)) -> sieve1(X)
activate1(n__nats1(X)) -> nats1(X)
activate1(X) -> X

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

filter3(cons2(X, Y), 0, M) -> cons2(0, n__filter3(activate1(Y), M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, n__filter3(activate1(Y), N, M))
sieve1(cons2(0, Y)) -> cons2(0, n__sieve1(activate1(Y)))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), n__sieve1(filter3(activate1(Y), N, N)))
nats1(N) -> cons2(N, n__nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))
filter3(X1, X2, X3) -> n__filter3(X1, X2, X3)
sieve1(X) -> n__sieve1(X)
nats1(X) -> n__nats1(X)
activate1(n__filter3(X1, X2, X3)) -> filter3(X1, X2, X3)
activate1(n__sieve1(X)) -> sieve1(X)
activate1(n__nats1(X)) -> nats1(X)
activate1(X) -> 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:

ACTIVATE1(n__nats1(X)) -> NATS1(X)
SIEVE1(cons2(s1(N), Y)) -> ACTIVATE1(Y)
FILTER3(cons2(X, Y), s1(N), M) -> ACTIVATE1(Y)
ZPRIMES -> SIEVE1(nats1(s1(s1(0))))
SIEVE1(cons2(0, Y)) -> ACTIVATE1(Y)
SIEVE1(cons2(s1(N), Y)) -> FILTER3(activate1(Y), N, N)
ZPRIMES -> NATS1(s1(s1(0)))
FILTER3(cons2(X, Y), 0, M) -> ACTIVATE1(Y)
ACTIVATE1(n__filter3(X1, X2, X3)) -> FILTER3(X1, X2, X3)
ACTIVATE1(n__sieve1(X)) -> SIEVE1(X)

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, n__filter3(activate1(Y), M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, n__filter3(activate1(Y), N, M))
sieve1(cons2(0, Y)) -> cons2(0, n__sieve1(activate1(Y)))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), n__sieve1(filter3(activate1(Y), N, N)))
nats1(N) -> cons2(N, n__nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))
filter3(X1, X2, X3) -> n__filter3(X1, X2, X3)
sieve1(X) -> n__sieve1(X)
nats1(X) -> n__nats1(X)
activate1(n__filter3(X1, X2, X3)) -> filter3(X1, X2, X3)
activate1(n__sieve1(X)) -> sieve1(X)
activate1(n__nats1(X)) -> nats1(X)
activate1(X) -> 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:

ACTIVATE1(n__nats1(X)) -> NATS1(X)
SIEVE1(cons2(s1(N), Y)) -> ACTIVATE1(Y)
FILTER3(cons2(X, Y), s1(N), M) -> ACTIVATE1(Y)
ZPRIMES -> SIEVE1(nats1(s1(s1(0))))
SIEVE1(cons2(0, Y)) -> ACTIVATE1(Y)
SIEVE1(cons2(s1(N), Y)) -> FILTER3(activate1(Y), N, N)
ZPRIMES -> NATS1(s1(s1(0)))
FILTER3(cons2(X, Y), 0, M) -> ACTIVATE1(Y)
ACTIVATE1(n__filter3(X1, X2, X3)) -> FILTER3(X1, X2, X3)
ACTIVATE1(n__sieve1(X)) -> SIEVE1(X)

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, n__filter3(activate1(Y), M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, n__filter3(activate1(Y), N, M))
sieve1(cons2(0, Y)) -> cons2(0, n__sieve1(activate1(Y)))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), n__sieve1(filter3(activate1(Y), N, N)))
nats1(N) -> cons2(N, n__nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))
filter3(X1, X2, X3) -> n__filter3(X1, X2, X3)
sieve1(X) -> n__sieve1(X)
nats1(X) -> n__nats1(X)
activate1(n__filter3(X1, X2, X3)) -> filter3(X1, X2, X3)
activate1(n__sieve1(X)) -> sieve1(X)
activate1(n__nats1(X)) -> nats1(X)
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 3 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP
          ↳ QDPOrderProof

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

SIEVE1(cons2(s1(N), Y)) -> ACTIVATE1(Y)
FILTER3(cons2(X, Y), s1(N), M) -> ACTIVATE1(Y)
SIEVE1(cons2(0, Y)) -> ACTIVATE1(Y)
SIEVE1(cons2(s1(N), Y)) -> FILTER3(activate1(Y), N, N)
FILTER3(cons2(X, Y), 0, M) -> ACTIVATE1(Y)
ACTIVATE1(n__filter3(X1, X2, X3)) -> FILTER3(X1, X2, X3)
ACTIVATE1(n__sieve1(X)) -> SIEVE1(X)

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, n__filter3(activate1(Y), M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, n__filter3(activate1(Y), N, M))
sieve1(cons2(0, Y)) -> cons2(0, n__sieve1(activate1(Y)))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), n__sieve1(filter3(activate1(Y), N, N)))
nats1(N) -> cons2(N, n__nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))
filter3(X1, X2, X3) -> n__filter3(X1, X2, X3)
sieve1(X) -> n__sieve1(X)
nats1(X) -> n__nats1(X)
activate1(n__filter3(X1, X2, X3)) -> filter3(X1, X2, X3)
activate1(n__sieve1(X)) -> sieve1(X)
activate1(n__nats1(X)) -> nats1(X)
activate1(X) -> 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(cons2(s1(N), Y)) -> ACTIVATE1(Y)
SIEVE1(cons2(0, Y)) -> ACTIVATE1(Y)
SIEVE1(cons2(s1(N), Y)) -> FILTER3(activate1(Y), N, N)
ACTIVATE1(n__sieve1(X)) -> SIEVE1(X)
The remaining pairs can at least be oriented weakly.

FILTER3(cons2(X, Y), s1(N), M) -> ACTIVATE1(Y)
FILTER3(cons2(X, Y), 0, M) -> ACTIVATE1(Y)
ACTIVATE1(n__filter3(X1, X2, X3)) -> FILTER3(X1, X2, X3)
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(ACTIVATE1(x1)) = 2·x1   
POL(FILTER3(x1, x2, x3)) = 2·x1   
POL(SIEVE1(x1)) = 2 + 2·x1   
POL(activate1(x1)) = x1   
POL(cons2(x1, x2)) = x2   
POL(filter3(x1, x2, x3)) = x1   
POL(n__filter3(x1, x2, x3)) = x1   
POL(n__nats1(x1)) = 0   
POL(n__sieve1(x1)) = 2 + 2·x1   
POL(nats1(x1)) = 0   
POL(s1(x1)) = 0   
POL(sieve1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented:

activate1(X) -> X
sieve1(cons2(0, Y)) -> cons2(0, n__sieve1(activate1(Y)))
filter3(X1, X2, X3) -> n__filter3(X1, X2, X3)
filter3(cons2(X, Y), s1(N), M) -> cons2(X, n__filter3(activate1(Y), N, M))
activate1(n__filter3(X1, X2, X3)) -> filter3(X1, X2, X3)
sieve1(X) -> n__sieve1(X)
nats1(N) -> cons2(N, n__nats1(s1(N)))
activate1(n__sieve1(X)) -> sieve1(X)
filter3(cons2(X, Y), 0, M) -> cons2(0, n__filter3(activate1(Y), M, M))
activate1(n__nats1(X)) -> nats1(X)
nats1(X) -> n__nats1(X)
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), n__sieve1(filter3(activate1(Y), N, N)))



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

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

FILTER3(cons2(X, Y), s1(N), M) -> ACTIVATE1(Y)
FILTER3(cons2(X, Y), 0, M) -> ACTIVATE1(Y)
ACTIVATE1(n__filter3(X1, X2, X3)) -> FILTER3(X1, X2, X3)

The TRS R consists of the following rules:

filter3(cons2(X, Y), 0, M) -> cons2(0, n__filter3(activate1(Y), M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, n__filter3(activate1(Y), N, M))
sieve1(cons2(0, Y)) -> cons2(0, n__sieve1(activate1(Y)))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), n__sieve1(filter3(activate1(Y), N, N)))
nats1(N) -> cons2(N, n__nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))
filter3(X1, X2, X3) -> n__filter3(X1, X2, X3)
sieve1(X) -> n__sieve1(X)
nats1(X) -> n__nats1(X)
activate1(n__filter3(X1, X2, X3)) -> filter3(X1, X2, X3)
activate1(n__sieve1(X)) -> sieve1(X)
activate1(n__nats1(X)) -> nats1(X)
activate1(X) -> 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.


FILTER3(cons2(X, Y), s1(N), M) -> ACTIVATE1(Y)
FILTER3(cons2(X, Y), 0, M) -> ACTIVATE1(Y)
ACTIVATE1(n__filter3(X1, X2, X3)) -> FILTER3(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(0) = 1   
POL(ACTIVATE1(x1)) = x1   
POL(FILTER3(x1, x2, x3)) = 2·x1 + x2   
POL(cons2(x1, x2)) = x2   
POL(n__filter3(x1, x2, x3)) = 1 + 2·x1 + x2 + 2·x3   
POL(s1(x1)) = 1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ 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, n__filter3(activate1(Y), M, M))
filter3(cons2(X, Y), s1(N), M) -> cons2(X, n__filter3(activate1(Y), N, M))
sieve1(cons2(0, Y)) -> cons2(0, n__sieve1(activate1(Y)))
sieve1(cons2(s1(N), Y)) -> cons2(s1(N), n__sieve1(filter3(activate1(Y), N, N)))
nats1(N) -> cons2(N, n__nats1(s1(N)))
zprimes -> sieve1(nats1(s1(s1(0))))
filter3(X1, X2, X3) -> n__filter3(X1, X2, X3)
sieve1(X) -> n__sieve1(X)
nats1(X) -> n__nats1(X)
activate1(n__filter3(X1, X2, X3)) -> filter3(X1, X2, X3)
activate1(n__sieve1(X)) -> sieve1(X)
activate1(n__nats1(X)) -> nats1(X)
activate1(X) -> 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.