Termination w.r.t. Q proof of TRS/TRCSRExIntrod_GM99

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

primes -> sieve₁(from₁(s₁(s₁(0))))
from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
head₁(cons₂(X, Y)) -> X
tail₁(cons₂(X, Y)) -> activate₁(Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
filter₂(s₁(s₁(X)), cons₂(Y, Z)) -> if₃(divides₂(s₁(s₁(X)), Y), n__filter₂(n__s₁(n__s₁(X)), activate₁(Z)), n__cons₂(Y, n__filter₂(X, n__sieve₁(Y))))
sieve₁(cons₂(X, Y)) -> cons₂(X, n__filter₂(X, n__sieve₁(activate₁(Y))))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
filter₂(X1, X2) -> n__filter₂(X1, X2)
cons₂(X1, X2) -> n__cons₂(X1, X2)
sieve₁(X) -> n__sieve₁(X)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__filter₂(X1, X2)) -> filter₂(activate₁(X1), activate₁(X2))
activate₁(n__cons₂(X1, X2)) -> cons₂(activate₁(X1), X2)
activate₁(n__sieve₁(X)) -> sieve₁(activate₁(X))
activate₁(X) -> X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

primes -> sieve₁(from₁(s₁(s₁(0))))
from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
head₁(cons₂(X, Y)) -> X
tail₁(cons₂(X, Y)) -> activate₁(Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
filter₂(s₁(s₁(X)), cons₂(Y, Z)) -> if₃(divides₂(s₁(s₁(X)), Y), n__filter₂(n__s₁(n__s₁(X)), activate₁(Z)), n__cons₂(Y, n__filter₂(X, n__sieve₁(Y))))
sieve₁(cons₂(X, Y)) -> cons₂(X, n__filter₂(X, n__sieve₁(activate₁(Y))))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
filter₂(X1, X2) -> n__filter₂(X1, X2)
cons₂(X1, X2) -> n__cons₂(X1, X2)
sieve₁(X) -> n__sieve₁(X)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__filter₂(X1, X2)) -> filter₂(activate₁(X1), activate₁(X2))
activate₁(n__cons₂(X1, X2)) -> cons₂(activate₁(X1), X2)
activate₁(n__sieve₁(X)) -> sieve₁(activate₁(X))
activate₁(X) -> X

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:

PRIMES -> S₁(0)
ACTIVATE₁(n__cons₂(X1, X2)) -> ACTIVATE₁(X1)
PRIMES -> S₁(s₁(0))
ACTIVATE₁(n__filter₂(X1, X2)) -> ACTIVATE₁(X2)
ACTIVATE₁(n__filter₂(X1, X2)) -> FILTER₂(activate₁(X1), activate₁(X2))
IF₃(true, X, Y) -> ACTIVATE₁(X)
FILTER₂(s₁(s₁(X)), cons₂(Y, Z)) -> IF₃(divides₂(s₁(s₁(X)), Y), n__filter₂(n__s₁(n__s₁(X)), activate₁(Z)), n__cons₂(Y, n__filter₂(X, n__sieve₁(Y))))
ACTIVATE₁(n__from₁(X)) -> FROM₁(activate₁(X))
ACTIVATE₁(n__s₁(X)) -> S₁(activate₁(X))
PRIMES -> SIEVE₁(from₁(s₁(s₁(0))))
ACTIVATE₁(n__filter₂(X1, X2)) -> ACTIVATE₁(X1)
SIEVE₁(cons₂(X, Y)) -> ACTIVATE₁(Y)
ACTIVATE₁(n__sieve₁(X)) -> ACTIVATE₁(X)
PRIMES -> FROM₁(s₁(s₁(0)))
FILTER₂(s₁(s₁(X)), cons₂(Y, Z)) -> ACTIVATE₁(Z)
SIEVE₁(cons₂(X, Y)) -> CONS₂(X, n__filter₂(X, n__sieve₁(activate₁(Y))))
IF₃(false, X, Y) -> ACTIVATE₁(Y)
FROM₁(X) -> CONS₂(X, n__from₁(n__s₁(X)))
ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__sieve₁(X)) -> SIEVE₁(activate₁(X))
ACTIVATE₁(n__cons₂(X1, X2)) -> CONS₂(activate₁(X1), X2)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)
TAIL₁(cons₂(X, Y)) -> ACTIVATE₁(Y)

The TRS R consists of the following rules:

primes -> sieve₁(from₁(s₁(s₁(0))))
from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
head₁(cons₂(X, Y)) -> X
tail₁(cons₂(X, Y)) -> activate₁(Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
filter₂(s₁(s₁(X)), cons₂(Y, Z)) -> if₃(divides₂(s₁(s₁(X)), Y), n__filter₂(n__s₁(n__s₁(X)), activate₁(Z)), n__cons₂(Y, n__filter₂(X, n__sieve₁(Y))))
sieve₁(cons₂(X, Y)) -> cons₂(X, n__filter₂(X, n__sieve₁(activate₁(Y))))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
filter₂(X1, X2) -> n__filter₂(X1, X2)
cons₂(X1, X2) -> n__cons₂(X1, X2)
sieve₁(X) -> n__sieve₁(X)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__filter₂(X1, X2)) -> filter₂(activate₁(X1), activate₁(X2))
activate₁(n__cons₂(X1, X2)) -> cons₂(activate₁(X1), X2)
activate₁(n__sieve₁(X)) -> sieve₁(activate₁(X))
activate₁(X) -> X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

PRIMES -> S₁(0)
ACTIVATE₁(n__cons₂(X1, X2)) -> ACTIVATE₁(X1)
PRIMES -> S₁(s₁(0))
ACTIVATE₁(n__filter₂(X1, X2)) -> ACTIVATE₁(X2)
ACTIVATE₁(n__filter₂(X1, X2)) -> FILTER₂(activate₁(X1), activate₁(X2))
IF₃(true, X, Y) -> ACTIVATE₁(X)
FILTER₂(s₁(s₁(X)), cons₂(Y, Z)) -> IF₃(divides₂(s₁(s₁(X)), Y), n__filter₂(n__s₁(n__s₁(X)), activate₁(Z)), n__cons₂(Y, n__filter₂(X, n__sieve₁(Y))))
ACTIVATE₁(n__from₁(X)) -> FROM₁(activate₁(X))
ACTIVATE₁(n__s₁(X)) -> S₁(activate₁(X))
PRIMES -> SIEVE₁(from₁(s₁(s₁(0))))
ACTIVATE₁(n__filter₂(X1, X2)) -> ACTIVATE₁(X1)
SIEVE₁(cons₂(X, Y)) -> ACTIVATE₁(Y)
ACTIVATE₁(n__sieve₁(X)) -> ACTIVATE₁(X)
PRIMES -> FROM₁(s₁(s₁(0)))
FILTER₂(s₁(s₁(X)), cons₂(Y, Z)) -> ACTIVATE₁(Z)
SIEVE₁(cons₂(X, Y)) -> CONS₂(X, n__filter₂(X, n__sieve₁(activate₁(Y))))
IF₃(false, X, Y) -> ACTIVATE₁(Y)
FROM₁(X) -> CONS₂(X, n__from₁(n__s₁(X)))
ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__sieve₁(X)) -> SIEVE₁(activate₁(X))
ACTIVATE₁(n__cons₂(X1, X2)) -> CONS₂(activate₁(X1), X2)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)
TAIL₁(cons₂(X, Y)) -> ACTIVATE₁(Y)

The TRS R consists of the following rules:

primes -> sieve₁(from₁(s₁(s₁(0))))
from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
head₁(cons₂(X, Y)) -> X
tail₁(cons₂(X, Y)) -> activate₁(Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
filter₂(s₁(s₁(X)), cons₂(Y, Z)) -> if₃(divides₂(s₁(s₁(X)), Y), n__filter₂(n__s₁(n__s₁(X)), activate₁(Z)), n__cons₂(Y, n__filter₂(X, n__sieve₁(Y))))
sieve₁(cons₂(X, Y)) -> cons₂(X, n__filter₂(X, n__sieve₁(activate₁(Y))))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
filter₂(X1, X2) -> n__filter₂(X1, X2)
cons₂(X1, X2) -> n__cons₂(X1, X2)
sieve₁(X) -> n__sieve₁(X)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__filter₂(X1, X2)) -> filter₂(activate₁(X1), activate₁(X2))
activate₁(n__cons₂(X1, X2)) -> cons₂(activate₁(X1), X2)
activate₁(n__sieve₁(X)) -> sieve₁(activate₁(X))
activate₁(X) -> X

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 13 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

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

ACTIVATE₁(n__cons₂(X1, X2)) -> ACTIVATE₁(X1)
ACTIVATE₁(n__filter₂(X1, X2)) -> ACTIVATE₁(X2)
SIEVE₁(cons₂(X, Y)) -> ACTIVATE₁(Y)
ACTIVATE₁(n__filter₂(X1, X2)) -> ACTIVATE₁(X1)
ACTIVATE₁(n__filter₂(X1, X2)) -> FILTER₂(activate₁(X1), activate₁(X2))
ACTIVATE₁(n__sieve₁(X)) -> ACTIVATE₁(X)
FILTER₂(s₁(s₁(X)), cons₂(Y, Z)) -> ACTIVATE₁(Z)
ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__sieve₁(X)) -> SIEVE₁(activate₁(X))
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

primes -> sieve₁(from₁(s₁(s₁(0))))
from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
head₁(cons₂(X, Y)) -> X
tail₁(cons₂(X, Y)) -> activate₁(Y)
if₃(true, X, Y) -> activate₁(X)
if₃(false, X, Y) -> activate₁(Y)
filter₂(s₁(s₁(X)), cons₂(Y, Z)) -> if₃(divides₂(s₁(s₁(X)), Y), n__filter₂(n__s₁(n__s₁(X)), activate₁(Z)), n__cons₂(Y, n__filter₂(X, n__sieve₁(Y))))
sieve₁(cons₂(X, Y)) -> cons₂(X, n__filter₂(X, n__sieve₁(activate₁(Y))))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
filter₂(X1, X2) -> n__filter₂(X1, X2)
cons₂(X1, X2) -> n__cons₂(X1, X2)
sieve₁(X) -> n__sieve₁(X)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__filter₂(X1, X2)) -> filter₂(activate₁(X1), activate₁(X2))
activate₁(n__cons₂(X1, X2)) -> cons₂(activate₁(X1), X2)
activate₁(n__sieve₁(X)) -> sieve₁(activate₁(X))
activate₁(X) -> X

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