Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 QDPOrderProof (⇔)
↳8 QDP
↳9 QDPOrderProof (⇔)
↳10 QDP
↳11 DependencyGraphProof (⇔)
↳12 QDP
↳13 QDPOrderProof (⇔)
↳14 QDP
↳15 QDPOrderProof (⇔)
↳16 QDP
↳17 DependencyGraphProof (⇔)
↳18 QDP
↳19 UsableRulesProof (⇔)
↳20 QDP
↳21 QDPSizeChangeProof (⇔)
↳22 TRUE

(0) Obligation:

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

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

FILTER(cons(X, Y), 0, M) → ACTIVATE(Y)
FILTER(cons(X, Y), s(N), M) → ACTIVATE(Y)
SIEVE(cons(0, Y)) → ACTIVATE(Y)
SIEVE(cons(s(N), Y)) → ACTIVATE(Y)
ZPRIMES → SIEVE(nats(s(s(0))))
ZPRIMES → NATS(s(s(0)))
ZPRIMES → S(s(0))
ZPRIMES → S(0)
ACTIVATE(n__filter(X1, X2, X3)) → FILTER(activate(X1), activate(X2), activate(X3))
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X3)
ACTIVATE(n__sieve(X)) → SIEVE(activate(X))
ACTIVATE(n__sieve(X)) → ACTIVATE(X)
ACTIVATE(n__nats(X)) → NATS(activate(X))
ACTIVATE(n__nats(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.

(4) Obligation:

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

ACTIVATE(n__filter(X1, X2, X3)) → FILTER(activate(X1), activate(X2), activate(X3))
FILTER(cons(X, Y), 0, M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X3)
ACTIVATE(n__sieve(X)) → SIEVE(activate(X))
SIEVE(cons(0, Y)) → ACTIVATE(Y)
ACTIVATE(n__sieve(X)) → ACTIVATE(X)
ACTIVATE(n__nats(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
SIEVE(cons(s(N), Y)) → ACTIVATE(Y)
FILTER(cons(X, Y), s(N), M) → ACTIVATE(Y)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__nats(X)) → ACTIVATE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x1)) = 0A + 0A·x1

POL(n__filter(x1, x2, x3)) = 0A + 0A·x1 + 0A·x2 + 0A·x3

POL(FILTER(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + 0A·x3

POL(activate(x1)) = -I + 0A·x1

POL(cons(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(0) = 3A

POL(n__sieve(x1)) = -I + 0A·x1

POL(SIEVE(x1)) = -I + 0A·x1

POL(n__nats(x1)) = 1A + 1A·x1

POL(n__s(x1)) = 0A + 0A·x1

POL(s(x1)) = 0A + 0A·x1

POL(filter(x1, x2, x3)) = 0A + 0A·x1 + 0A·x2 + 0A·x3

POL(nats(x1)) = 1A + 1A·x1

POL(sieve(x1)) = -I + 0A·x1

The following usable rules [FROCOS05] were oriented:

filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
s(X) → n__s(X)
nats(X) → n__nats(X)
sieve(X) → n__sieve(X)
filter(X1, X2, X3) → n__filter(X1, X2, X3)
nats(N) → cons(N, n__nats(n__s(N)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
activate(X) → X
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))

(6) Obligation:

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

ACTIVATE(n__filter(X1, X2, X3)) → FILTER(activate(X1), activate(X2), activate(X3))
FILTER(cons(X, Y), 0, M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X3)
ACTIVATE(n__sieve(X)) → SIEVE(activate(X))
SIEVE(cons(0, Y)) → ACTIVATE(Y)
ACTIVATE(n__sieve(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
SIEVE(cons(s(N), Y)) → ACTIVATE(Y)
FILTER(cons(X, Y), s(N), M) → ACTIVATE(Y)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__sieve(X)) → ACTIVATE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x1)) = -I + 0A·x1

POL(n__filter(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + 0A·x3

POL(FILTER(x1, x2, x3)) = -I + 0A·x1 + -I·x2 + -I·x3

POL(activate(x1)) = -I + 0A·x1

POL(cons(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(0) = 0A

POL(n__sieve(x1)) = 0A + 1A·x1

POL(SIEVE(x1)) = 0A + 0A·x1

POL(n__s(x1)) = -I + 0A·x1

POL(s(x1)) = -I + 0A·x1

POL(filter(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + 0A·x3

POL(nats(x1)) = 3A + 0A·x1

POL(n__nats(x1)) = 3A + 0A·x1

POL(sieve(x1)) = 0A + 1A·x1

The following usable rules [FROCOS05] were oriented:

filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
s(X) → n__s(X)
nats(X) → n__nats(X)
sieve(X) → n__sieve(X)
filter(X1, X2, X3) → n__filter(X1, X2, X3)
nats(N) → cons(N, n__nats(n__s(N)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
activate(X) → X
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))

(8) Obligation:

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

ACTIVATE(n__filter(X1, X2, X3)) → FILTER(activate(X1), activate(X2), activate(X3))
FILTER(cons(X, Y), 0, M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X3)
ACTIVATE(n__sieve(X)) → SIEVE(activate(X))
SIEVE(cons(0, Y)) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)
SIEVE(cons(s(N), Y)) → ACTIVATE(Y)
FILTER(cons(X, Y), s(N), M) → ACTIVATE(Y)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__sieve(X)) → SIEVE(activate(X))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x1)) = 0A + 0A·x1

POL(n__filter(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + 0A·x3

POL(FILTER(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + 0A·x3

POL(activate(x1)) = 0A + 0A·x1

POL(cons(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(0) = 0A

POL(n__sieve(x1)) = 1A + 1A·x1

POL(SIEVE(x1)) = -I + 0A·x1

POL(n__s(x1)) = -I + 0A·x1

POL(s(x1)) = -I + 0A·x1

POL(filter(x1, x2, x3)) = 0A + 0A·x1 + 0A·x2 + 0A·x3

POL(nats(x1)) = 0A + 0A·x1

POL(n__nats(x1)) = 0A + 0A·x1

POL(sieve(x1)) = 1A + 1A·x1

The following usable rules [FROCOS05] were oriented:

filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
s(X) → n__s(X)
nats(X) → n__nats(X)
sieve(X) → n__sieve(X)
filter(X1, X2, X3) → n__filter(X1, X2, X3)
nats(N) → cons(N, n__nats(n__s(N)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
activate(X) → X
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))

(10) Obligation:

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

ACTIVATE(n__filter(X1, X2, X3)) → FILTER(activate(X1), activate(X2), activate(X3))
FILTER(cons(X, Y), 0, M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X3)
SIEVE(cons(0, Y)) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)
SIEVE(cons(s(N), Y)) → ACTIVATE(Y)
FILTER(cons(X, Y), s(N), M) → ACTIVATE(Y)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(12) Obligation:

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

FILTER(cons(X, Y), 0, M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → FILTER(activate(X1), activate(X2), activate(X3))
FILTER(cons(X, Y), s(N), M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X3)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(FILTER(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + 1A·x3

POL(cons(x1, x2)) = -I + -I·x1 + 0A·x2

POL(0) = 2A

POL(ACTIVATE(x1)) = 0A + 0A·x1

POL(n__filter(x1, x2, x3)) = 4A + 0A·x1 + 0A·x2 + 1A·x3

POL(activate(x1)) = -I + 0A·x1

POL(s(x1)) = 0A + 0A·x1

POL(n__s(x1)) = 0A + 0A·x1

POL(filter(x1, x2, x3)) = 4A + 0A·x1 + 0A·x2 + 1A·x3

POL(nats(x1)) = 0A + -I·x1

POL(n__nats(x1)) = 0A + -I·x1

POL(sieve(x1)) = 0A + -I·x1

POL(n__sieve(x1)) = 0A + -I·x1

The following usable rules [FROCOS05] were oriented:

filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
s(X) → n__s(X)
nats(X) → n__nats(X)
sieve(X) → n__sieve(X)
filter(X1, X2, X3) → n__filter(X1, X2, X3)
nats(N) → cons(N, n__nats(n__s(N)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
activate(X) → X
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))

(14) Obligation:

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

FILTER(cons(X, Y), 0, M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → FILTER(activate(X1), activate(X2), activate(X3))
FILTER(cons(X, Y), s(N), M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__filter(X1, X2, X3)) → FILTER(activate(X1), activate(X2), activate(X3))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(FILTER(x1, x2, x3)) = -I + 0A·x1 + -I·x2 + 2A·x3

POL(cons(x1, x2)) = 0A + -I·x1 + 0A·x2

POL(0) = 0A

POL(ACTIVATE(x1)) = 0A + 0A·x1

POL(n__filter(x1, x2, x3)) = 0A + 1A·x1 + 0A·x2 + 5A·x3

POL(activate(x1)) = -I + 0A·x1

POL(s(x1)) = -I + 5A·x1

POL(n__s(x1)) = -I + 5A·x1

POL(filter(x1, x2, x3)) = 0A + 1A·x1 + 0A·x2 + 5A·x3

POL(nats(x1)) = 0A + -I·x1

POL(n__nats(x1)) = 0A + -I·x1

POL(sieve(x1)) = 0A + -I·x1

POL(n__sieve(x1)) = 0A + -I·x1

The following usable rules [FROCOS05] were oriented:

filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
s(X) → n__s(X)
nats(X) → n__nats(X)
sieve(X) → n__sieve(X)
filter(X1, X2, X3) → n__filter(X1, X2, X3)
nats(N) → cons(N, n__nats(n__s(N)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
activate(X) → X
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))

(16) Obligation:

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

FILTER(cons(X, Y), 0, M) → ACTIVATE(Y)
FILTER(cons(X, Y), s(N), M) → ACTIVATE(Y)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(18) Obligation:

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

ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

filter(cons(X, Y), 0, M) → cons(0, n__filter(activate(Y), M, M))
filter(cons(X, Y), s(N), M) → cons(X, n__filter(activate(Y), N, M))
sieve(cons(0, Y)) → cons(0, n__sieve(activate(Y)))
sieve(cons(s(N), Y)) → cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
nats(N) → cons(N, n__nats(n__s(N)))
zprimes → sieve(nats(s(s(0))))
filter(X1, X2, X3) → n__filter(X1, X2, X3)
sieve(X) → n__sieve(X)
nats(X) → n__nats(X)
s(X) → n__s(X)
activate(n__filter(X1, X2, X3)) → filter(activate(X1), activate(X2), activate(X3))
activate(n__sieve(X)) → sieve(activate(X))
activate(n__nats(X)) → nats(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

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

(19) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(20) Obligation:

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

ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X2)
    The graph contains the following edges 1 > 1

  • ACTIVATE(n__filter(X1, X2, X3)) → ACTIVATE(X1)
    The graph contains the following edges 1 > 1

  • ACTIVATE(n__s(X)) → ACTIVATE(X)
    The graph contains the following edges 1 > 1

(22) TRUE