YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex9_BLR02_FR.ari

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

0 QTRS
↳1 DependencyPairsProof (⇔, 0 ms)
↳2 QDP
↳3 DependencyGraphProof (⇔, 0 ms)
↳4 QDP
↳5 QDPOrderProof (⇔, 83 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 35 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 69 ms)
↳12 QDP
↳13 DependencyGraphProof (⇔, 0 ms)
↳14 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,JAR06].


The following pairs can be oriented strictly and are deleted.


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

POL(ACTIVATE(x1)) = 2A + 0A·x1

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

POL(FILTER(x1, x2, x3)) = 2A + 0A·x1 + 1A·x2 + -I·x3

POL(activate(x1)) = -I + 0A·x1

POL(cons(x1, x2)) = 0A + 1A·x1 + 0A·x2

POL(0) = 0A

POL(n__sieve(x1)) = 2A + 2A·x1

POL(SIEVE(x1)) = 2A + 1A·x1

POL(n__nats(x1)) = 5A + 1A·x1

POL(n__s(x1)) = 4A + 0A·x1

POL(s(x1)) = 4A + 0A·x1

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

POL(sieve(x1)) = 2A + 2A·x1

POL(nats(x1)) = 5A + 1A·x1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(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)
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,JAR06].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__sieve(X)) → SIEVE(activate(X))
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)) = 5A + 2A·x1

POL(n__filter(x1, x2, x3)) = 1A + 0A·x1 + 2A·x2 + 2A·x3

POL(FILTER(x1, x2, x3)) = 5A + 2A·x1 + -I·x2 + -I·x3

POL(activate(x1)) = -I + 0A·x1

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

POL(0) = 0A

POL(n__sieve(x1)) = 5A + 4A·x1

POL(SIEVE(x1)) = 5A + 2A·x1

POL(n__s(x1)) = -I + 0A·x1

POL(s(x1)) = -I + 0A·x1

POL(filter(x1, x2, x3)) = 1A + 0A·x1 + 2A·x2 + 2A·x3

POL(sieve(x1)) = 5A + 4A·x1

POL(n__nats(x1)) = -I + 3A·x1

POL(nats(x1)) = -I + 3A·x1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(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)
SIEVE(cons(0, Y)) → ACTIVATE(Y)
ACTIVATE(n__s(X)) → ACTIVATE(X)
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) DependencyGraphProof (EQUIVALENT transformation)

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

(10) 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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


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 remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO,RATPOLO]:

POL(0) = 0   
POL(ACTIVATE(x1)) = [1/2]x1   
POL(FILTER(x1, x2, x3)) = x1 + [1/4]x2 + [1/4]x3   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = [1/2]x2   
POL(filter(x1, x2, x3)) = [1] + [2]x1 + x2 + x3   
POL(n__filter(x1, x2, x3)) = [1] + [2]x1 + x2 + x3   
POL(n__nats(x1)) = 0   
POL(n__s(x1)) = [1/4] + x1   
POL(n__sieve(x1)) = 0   
POL(nats(x1)) = 0   
POL(s(x1)) = [1/4] + x1   
POL(sieve(x1)) = 0   
The value of delta used in the strict ordering is 1/16.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(12) Obligation:

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

FILTER(cons(X, Y), 0, 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.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE