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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 QDPOrderProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 QDPOrderProof (⇔)
↳8 QDP
↳9 QDPOrderProof (⇔)
↳10 QDP
↳11 QDPOrderProof (⇔)
↳12 QDP
↳13 QDPOrderProof (⇔)
↳14 QDP
↳15 PisEmptyProof (⇔)
↳16 TRUE

(0) Obligation:

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

a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__sieve(X) → sieve(X)
a__nats(X) → nats(X)
a__zprimes → zprimes

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:

A__FILTER(cons(X, Y), s(N), M) → MARK(X)
A__SIEVE(cons(s(N), Y)) → MARK(N)
A__NATS(N) → MARK(N)
A__ZPRIMES → A__SIEVE(a__nats(s(s(0))))
A__ZPRIMES → A__NATS(s(s(0)))
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(filter(X1, X2, X3)) → MARK(X1)
MARK(filter(X1, X2, X3)) → MARK(X2)
MARK(filter(X1, X2, X3)) → MARK(X3)
MARK(sieve(X)) → A__SIEVE(mark(X))
MARK(sieve(X)) → MARK(X)
MARK(nats(X)) → A__NATS(mark(X))
MARK(nats(X)) → MARK(X)
MARK(zprimes) → A__ZPRIMES
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__sieve(X) → sieve(X)
a__nats(X) → nats(X)
a__zprimes → zprimes

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__FILTER(cons(X, Y), s(N), M) → MARK(X)
A__ZPRIMES → A__SIEVE(a__nats(s(s(0))))
A__ZPRIMES → A__NATS(s(s(0)))
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(filter(X1, X2, X3)) → MARK(X1)
MARK(filter(X1, X2, X3)) → MARK(X2)
MARK(filter(X1, X2, X3)) → MARK(X3)
MARK(zprimes) → A__ZPRIMES
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__FILTER(x0, x1, x2, x3)  =  A__FILTER(x0)
MARK(x0, x1)  =  MARK(x0)
A__SIEVE(x0, x1)  =  A__SIEVE(x1)
A__NATS(x0, x1)  =  A__NATS(x0)
A__ZPRIMES(x0)  =  A__ZPRIMES(x0)

Tags:
A__FILTER has argument tags [10,8,2,15] and root tag 0
MARK has argument tags [4,12] and root tag 4
A__SIEVE has argument tags [15,4] and root tag 4
A__NATS has argument tags [4,11] and root tag 4
A__ZPRIMES has argument tags [4] and root tag 6

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__FILTER(x1, x2, x3)  =  A__FILTER(x1, x2, x3)
cons(x1, x2)  =  x1
s(x1)  =  x1
MARK(x1)  =  x1
A__SIEVE(x1)  =  A__SIEVE
A__NATS(x1)  =  x1
A__ZPRIMES  =  A__ZPRIMES
a__nats(x1)  =  x1
0  =  0
filter(x1, x2, x3)  =  filter(x1, x2, x3)
mark(x1)  =  x1
sieve(x1)  =  x1
nats(x1)  =  x1
zprimes  =  zprimes
a__filter(x1, x2, x3)  =  a__filter(x1, x2, x3)
a__sieve(x1)  =  x1
a__zprimes  =  a__zprimes

Lexicographic path order with status [LPO].
Quasi-Precedence:
ASIEVE > [AZPRIMES, 0]
[filter3, afilter3] > AFILTER3 > [AZPRIMES, 0]
[zprimes, azprimes] > [AZPRIMES, 0]

Status:
AFILTER3: [1,3,2]
ASIEVE: []
AZPRIMES: []
0: []
filter3: [2,3,1]
zprimes: []
afilter3: [2,3,1]
azprimes: []


The following usable rules [FROCOS05] were oriented:

a__nats(N) → cons(mark(N), nats(s(N)))
a__nats(X) → nats(X)
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__zprimes → a__sieve(a__nats(s(s(0))))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(X) → sieve(X)
a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__zprimes → zprimes

(4) Obligation:

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

A__SIEVE(cons(s(N), Y)) → MARK(N)
A__NATS(N) → MARK(N)
MARK(sieve(X)) → A__SIEVE(mark(X))
MARK(sieve(X)) → MARK(X)
MARK(nats(X)) → A__NATS(mark(X))
MARK(nats(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__sieve(X) → sieve(X)
a__nats(X) → nats(X)
a__zprimes → zprimes

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.


MARK(sieve(X)) → MARK(X)
MARK(nats(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__SIEVE(x0, x1)  =  A__SIEVE(x0)
MARK(x0, x1)  =  MARK(x0)
A__NATS(x0, x1)  =  A__NATS(x0)

Tags:
A__SIEVE has argument tags [2,0] and root tag 0
MARK has argument tags [2,2] and root tag 0
A__NATS has argument tags [2,5] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__SIEVE(x1)  =  A__SIEVE(x1)
cons(x1, x2)  =  x1
s(x1)  =  x1
MARK(x1)  =  MARK(x1)
A__NATS(x1)  =  A__NATS(x1)
sieve(x1)  =  sieve(x1)
mark(x1)  =  mark(x1)
nats(x1)  =  nats(x1)
filter(x1, x2, x3)  =  filter(x1, x2, x3)
a__filter(x1, x2, x3)  =  a__filter(x1, x2, x3)
a__sieve(x1)  =  a__sieve(x1)
a__nats(x1)  =  a__nats(x1)
zprimes  =  zprimes
a__zprimes  =  a__zprimes
0  =  0

Lexicographic path order with status [LPO].
Quasi-Precedence:
[ASIEVE1, MARK1, ANATS1] > [sieve1, mark1, nats1, filter3, afilter3, asieve1, anats1]
[zprimes, azprimes] > 0 > [sieve1, mark1, nats1, filter3, afilter3, asieve1, anats1]

Status:
ASIEVE1: [1]
MARK1: [1]
ANATS1: [1]
sieve1: [1]
mark1: [1]
nats1: [1]
filter3: [1,3,2]
afilter3: [1,3,2]
asieve1: [1]
anats1: [1]
zprimes: []
azprimes: []
0: []


The following usable rules [FROCOS05] were oriented:

mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(X) → sieve(X)
a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__nats(X) → nats(X)
a__zprimes → zprimes

(6) Obligation:

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

A__SIEVE(cons(s(N), Y)) → MARK(N)
A__NATS(N) → MARK(N)
MARK(sieve(X)) → A__SIEVE(mark(X))
MARK(nats(X)) → A__NATS(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__sieve(X) → sieve(X)
a__nats(X) → nats(X)
a__zprimes → zprimes

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.


A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(sieve(X)) → A__SIEVE(mark(X))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__SIEVE(x0, x1)  =  A__SIEVE(x0)
MARK(x0, x1)  =  MARK(x0)
A__NATS(x0, x1)  =  A__NATS(x1)

Tags:
A__SIEVE has argument tags [1,0] and root tag 1
MARK has argument tags [1,6] and root tag 0
A__NATS has argument tags [0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__SIEVE(x1)  =  x1
cons(x1, x2)  =  x1
s(x1)  =  x1
MARK(x1)  =  x1
A__NATS(x1)  =  A__NATS
sieve(x1)  =  sieve(x1)
mark(x1)  =  x1
nats(x1)  =  x1
filter(x1, x2, x3)  =  filter(x1, x2, x3)
a__filter(x1, x2, x3)  =  a__filter(x1, x2, x3)
a__sieve(x1)  =  a__sieve(x1)
a__nats(x1)  =  x1
zprimes  =  zprimes
a__zprimes  =  a__zprimes
0  =  0

Lexicographic path order with status [LPO].
Quasi-Precedence:
[zprimes, azprimes] > [sieve1, asieve1] > [filter3, afilter3] > 0

Status:
ANATS: []
sieve1: [1]
filter3: [2,3,1]
afilter3: [2,3,1]
asieve1: [1]
zprimes: []
azprimes: []
0: []


The following usable rules [FROCOS05] were oriented:

mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(X) → sieve(X)
a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__nats(X) → nats(X)
a__zprimes → zprimes

(8) Obligation:

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

A__NATS(N) → MARK(N)
MARK(nats(X)) → A__NATS(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__sieve(X) → sieve(X)
a__nats(X) → nats(X)
a__zprimes → zprimes

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.


MARK(s(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__NATS(x0, x1)  =  A__NATS(x1)
MARK(x0, x1)  =  MARK(x0)

Tags:
A__NATS has argument tags [1,3] and root tag 0
MARK has argument tags [3,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__NATS(x1)  =  A__NATS(x1)
MARK(x1)  =  x1
nats(x1)  =  nats(x1)
mark(x1)  =  mark(x1)
cons(x1, x2)  =  x1
s(x1)  =  s(x1)
filter(x1, x2, x3)  =  filter(x1)
a__filter(x1, x2, x3)  =  a__filter(x1)
sieve(x1)  =  sieve(x1)
a__sieve(x1)  =  a__sieve(x1)
a__nats(x1)  =  a__nats(x1)
zprimes  =  zprimes
a__zprimes  =  a__zprimes
0  =  0

Lexicographic path order with status [LPO].
Quasi-Precedence:
[zprimes, azprimes] > [nats1, mark1, filter1, afilter1, sieve1, asieve1, anats1, 0] > ANATS1
[zprimes, azprimes] > [nats1, mark1, filter1, afilter1, sieve1, asieve1, anats1, 0] > s1

Status:
ANATS1: [1]
nats1: [1]
mark1: [1]
s1: [1]
filter1: [1]
afilter1: [1]
sieve1: [1]
asieve1: [1]
anats1: [1]
zprimes: []
azprimes: []
0: []


The following usable rules [FROCOS05] were oriented:

mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(X) → sieve(X)
a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__nats(X) → nats(X)
a__zprimes → zprimes

(10) Obligation:

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

A__NATS(N) → MARK(N)
MARK(nats(X)) → A__NATS(mark(X))
MARK(cons(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__sieve(X) → sieve(X)
a__nats(X) → nats(X)
a__zprimes → zprimes

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__NATS(N) → MARK(N)
MARK(nats(X)) → A__NATS(mark(X))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__NATS(x0, x1)  =  A__NATS(x1)
MARK(x0, x1)  =  MARK(x0)

Tags:
A__NATS has argument tags [0,1] and root tag 1
MARK has argument tags [1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__NATS(x1)  =  A__NATS
MARK(x1)  =  x1
nats(x1)  =  nats(x1)
mark(x1)  =  x1
cons(x1, x2)  =  x1
filter(x1, x2, x3)  =  filter(x1, x2, x3)
a__filter(x1, x2, x3)  =  a__filter(x1, x2, x3)
sieve(x1)  =  sieve
a__sieve(x1)  =  a__sieve
a__nats(x1)  =  a__nats(x1)
zprimes  =  zprimes
a__zprimes  =  a__zprimes
0  =  0
s(x1)  =  s

Lexicographic path order with status [LPO].
Quasi-Precedence:
ANATS > 0
[nats1, anats1, zprimes, azprimes] > [sieve, asieve, s] > 0
[filter3, afilter3] > 0

Status:
ANATS: []
nats1: [1]
filter3: [2,3,1]
afilter3: [2,3,1]
sieve: []
asieve: []
anats1: [1]
zprimes: []
azprimes: []
0: []
s: []


The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

MARK(cons(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__sieve(X) → sieve(X)
a__nats(X) → nats(X)
a__zprimes → zprimes

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.


MARK(cons(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x1)

Tags:
MARK has argument tags [0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  MARK
cons(x1, x2)  =  cons(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
MARK: []
cons2: [1,2]


The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

a__filter(cons(X, Y), 0, M) → cons(0, filter(Y, M, M))
a__filter(cons(X, Y), s(N), M) → cons(mark(X), filter(Y, N, M))
a__sieve(cons(0, Y)) → cons(0, sieve(Y))
a__sieve(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
a__nats(N) → cons(mark(N), nats(s(N)))
a__zprimes → a__sieve(a__nats(s(s(0))))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) → a__sieve(mark(X))
mark(nats(X)) → a__nats(mark(X))
mark(zprimes) → a__zprimes
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
a__sieve(X) → sieve(X)
a__nats(X) → nats(X)
a__zprimes → zprimes

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

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE