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:

a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes

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:

A__ZPRIMES -> A__NATS1(s1(s1(0)))
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__NATS1(N) -> MARK1(N)
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
A__ZPRIMES -> A__SIEVE1(a__nats1(s1(s1(0))))
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
MARK1(zprimes) -> A__ZPRIMES
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(sieve1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ QDPOrderProof

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

A__ZPRIMES -> A__NATS1(s1(s1(0)))
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__NATS1(N) -> MARK1(N)
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
A__ZPRIMES -> A__SIEVE1(a__nats1(s1(s1(0))))
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
MARK1(zprimes) -> A__ZPRIMES
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(sieve1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes

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.


A__ZPRIMES -> A__NATS1(s1(s1(0)))
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
A__ZPRIMES -> A__SIEVE1(a__nats1(s1(s1(0))))
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
MARK1(zprimes) -> A__ZPRIMES
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
The remaining pairs can at least by weakly be oriented.

A__NATS1(N) -> MARK1(N)
MARK1(sieve1(X)) -> MARK1(X)
Used ordering: Combined order from the following AFS and order.
A__ZPRIMES  =  A__ZPRIMES
A__NATS1(x1)  =  A__NATS1(x1)
s1(x1)  =  s1(x1)
0  =  0
MARK1(x1)  =  MARK1(x1)
filter3(x1, x2, x3)  =  filter3(x1, x2, x3)
nats1(x1)  =  nats1(x1)
mark1(x1)  =  x1
A__FILTER3(x1, x2, x3)  =  x1
A__SIEVE1(x1)  =  x1
a__nats1(x1)  =  a__nats1(x1)
sieve1(x1)  =  x1
zprimes  =  zprimes
cons2(x1, x2)  =  cons1(x1)
a__zprimes  =  a__zprimes
a__filter3(x1, x2, x3)  =  a__filter3(x1, x2, x3)
a__sieve1(x1)  =  x1

Lexicographic Path Order [19].
Precedence:
[AZPRIMES, zprimes, azprimes] > 0 > [nats1, anats1, cons1] > [ANATS1, MARK1]
[AZPRIMES, zprimes, azprimes] > 0 > [nats1, anats1, cons1] > s1
[filter3, afilter3] > 0 > [nats1, anats1, cons1] > [ANATS1, MARK1]
[filter3, afilter3] > 0 > [nats1, anats1, cons1] > s1


The following usable rules [14] were oriented:

a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__nats1(X) -> nats1(X)
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
a__zprimes -> zprimes
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__sieve1(X) -> sieve1(X)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
QDP
          ↳ DependencyGraphProof

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

A__NATS1(N) -> MARK1(N)
MARK1(sieve1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes

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 1 less node.

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

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

MARK1(sieve1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes

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.


MARK1(sieve1(X)) -> MARK1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
MARK1(x1)  =  MARK1(x1)
sieve1(x1)  =  sieve1(x1)

Lexicographic Path Order [19].
Precedence:
sieve1 > MARK1


The following usable rules [14] were oriented: none



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

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

a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes

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.