Termination w.r.t. Q proof of /tmp/tmpanSm_h/Ex9_BLR02

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 PisEmptyProof (⇔)

↳12 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__SIEVE(cons(s(N), Y)) → 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(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(x1, x2, x3)
MARK(x0, x1) = MARK(x1)
A__SIEVE(x0, x1) = A__SIEVE(x1)
A__NATS(x0, x1) = A__NATS(x1)
A__ZPRIMES(x0) = A__ZPRIMES(x0)

Tags:
A__FILTER has argument tags [8,2,14,7] and root tag 1
MARK has argument tags [0,2] and root tag 0
A__SIEVE has argument tags [0,3] and root tag 5
A__NATS has argument tags [11,2] and root tag 0
A__ZPRIMES has argument tags [4] and root tag 1

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(x2, x3)
cons(x1, x2) = x1
s(x1) = x1
MARK(x1) = MARK(x1)
A__SIEVE(x1) = A__SIEVE
A__NATS(x1) = A__NATS(x1)
A__ZPRIMES = A__ZPRIMES
a__nats(x1) = x1
0 = 0
filter(x1, x2, x3) = filter(x1, x2, x3)
mark(x1) = x1
sieve(x1) = sieve(x1)
nats(x1) = x1
zprimes = zprimes
a__filter(x1, x2, x3) = a__filter(x1, x2, x3)
a__sieve(x1) = a__sieve(x1)
a__zprimes = a__zprimes

Recursive path order with status [RPO].
Quasi-Precedence:

A_{_SIEVE} > [MARK₁, A_{_NATS₁}, A_{_ZPRIMES}, 0]
[zprimes, a_{_zprimes}] > [A_{_FILTER₂}, filter₃, sieve₁, a_{_filter₃}, a_{_sieve₁}] > [MARK₁, A_{_NATS₁}, A_{_ZPRIMES}, 0]

Status:

A_{_FILTER₂}: multiset
MARK₁: multiset
A_{_SIEVE}: multiset
A_{_NATS₁}: [1]
A_{_ZPRIMES}: multiset
0: multiset
filter₃: multiset
sieve₁: [1]
zprimes: multiset
a_{_filter₃}: multiset
a_{_sieve₁}: [1]
a_{_zprimes}: multiset

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__NATS(N) → MARK(N)
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(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__NATS(x0, x1) = A__NATS(x0)
MARK(x0, x1) = MARK(x1)

Tags:
A__NATS has argument tags [0,1] and root tag 0
MARK has argument tags [2,0] 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) = x1
MARK(x1) = x1
nats(x1) = nats(x1)
mark(x1) = mark(x1)
cons(x1, x2) = x1
s(x1) = 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

Recursive path order with status [RPO].
Quasi-Precedence:

[zprimes, a_{_zprimes}] > [nats₁, mark₁, filter₁, a_{_filter₁}, sieve₁, a_{_sieve₁}, a_{_nats₁}, 0]

Status:

nats₁: multiset
mark₁: multiset
filter₁: multiset
a_{_filter₁}: multiset
sieve₁: multiset
a_{_sieve₁}: multiset
a_{_nats₁}: multiset
zprimes: multiset
a_{_zprimes}: multiset
0: multiset

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__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.

(7) 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(x0)
MARK(x0, x1) = MARK(x1)

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

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) = x1
MARK(x1) = MARK(x1)
nats(x1) = x1
mark(x1) = x1
cons(x1, x2) = x1
s(x1) = s(x1)
filter(x1, x2, x3) = filter(x1, x2, x3)
a__filter(x1, x2, x3) = a__filter(x1, x2, x3)
sieve(x1) = x1
a__sieve(x1) = x1
a__nats(x1) = x1
zprimes = zprimes
a__zprimes = a__zprimes
0 = 0

Recursive path order with status [RPO].
Quasi-Precedence:

[filter₃, a_{_filter₃}] > 0
[zprimes, a_{_zprimes}] > s₁
[zprimes, a_{_zprimes}] > 0

Status:

MARK₁: multiset
s₁: multiset
filter₃: [1,2,3]
a_{_filter₃}: [1,2,3]
zprimes: multiset
a_{_zprimes}: multiset
0: multiset

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)

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.

A__NATS(N) → MARK(N)
MARK(nats(X)) → A__NATS(mark(X))
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:
A__NATS(x0, x1) = A__NATS(x0, x1)
MARK(x0, x1) = MARK(x1)

Tags:
A__NATS has argument tags [2,3] and root tag 0
MARK has argument tags [0,2] and root tag 1

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) = MARK
nats(x1) = nats(x1)
mark(x1) = x1
cons(x1, x2) = cons(x1)
filter(x1, x2, x3) = x1
a__filter(x1, x2, x3) = x1
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

Recursive path order with status [RPO].
Quasi-Precedence:

MARK > [A_{_NATS₁}, nats₁, a_{_nats₁}] > [cons₁, s] > 0
[zprimes, a_{_zprimes}] > [A_{_NATS₁}, nats₁, a_{_nats₁}] > [cons₁, s] > 0
[zprimes, a_{_zprimes}] > [sieve, a_{_sieve}] > [cons₁, s] > 0

Status:

A_{_NATS₁}: multiset
MARK: []
nats₁: multiset
cons₁: [1]
sieve: []
a_{_sieve}: []
a_{_nats₁}: multiset
zprimes: multiset
a_{_zprimes}: multiset
0: multiset
s: multiset

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:
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.

(11) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) QDPOrderProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) PisEmptyProof (EQUIVALENT transformation)

(12) TRUE