a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
↳ QTRS
↳ DependencyPairsProof
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
A__ZPRIMES -> A__NATS1(s1(s1(0)))
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__NATS1(N) -> MARK1(N)
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
A__ZPRIMES -> A__SIEVE1(a__nats1(s1(s1(0))))
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
MARK1(zprimes) -> A__ZPRIMES
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(sieve1(X)) -> MARK1(X)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
A__ZPRIMES -> A__NATS1(s1(s1(0)))
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__NATS1(N) -> MARK1(N)
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
A__ZPRIMES -> A__SIEVE1(a__nats1(s1(s1(0))))
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
MARK1(zprimes) -> A__ZPRIMES
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(sieve1(X)) -> MARK1(X)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK1(zprimes) -> A__ZPRIMES
Used ordering: Polynomial interpretation [21]:
A__ZPRIMES -> A__NATS1(s1(s1(0)))
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__NATS1(N) -> MARK1(N)
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
A__ZPRIMES -> A__SIEVE1(a__nats1(s1(s1(0))))
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(sieve1(X)) -> MARK1(X)
POL(0) = 0
POL(A__FILTER3(x1, x2, x3)) = 2·x1
POL(A__NATS1(x1)) = 2·x1
POL(A__SIEVE1(x1)) = 2·x1
POL(A__ZPRIMES) = 0
POL(MARK1(x1)) = x1
POL(a__filter3(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(a__nats1(x1)) = 2·x1
POL(a__sieve1(x1)) = 2·x1
POL(a__zprimes) = 1
POL(cons2(x1, x2)) = 2·x1
POL(filter3(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(mark1(x1)) = x1
POL(nats1(x1)) = 2·x1
POL(s1(x1)) = 2·x1
POL(sieve1(x1)) = 2·x1
POL(zprimes) = 1
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__zprimes -> zprimes
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
a__nats1(X) -> nats1(X)
mark1(0) -> 0
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
a__sieve1(X) -> sieve1(X)
mark1(nats1(X)) -> a__nats1(mark1(X))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
mark1(zprimes) -> a__zprimes
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
A__ZPRIMES -> A__NATS1(s1(s1(0)))
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__NATS1(N) -> MARK1(N)
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
A__ZPRIMES -> A__SIEVE1(a__nats1(s1(s1(0))))
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(sieve1(X)) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__NATS1(N) -> MARK1(N)
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(sieve1(X)) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK1(nats1(X)) -> A__NATS1(mark1(X))
MARK1(nats1(X)) -> MARK1(X)
Used ordering: Polynomial interpretation [21]:
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__NATS1(N) -> MARK1(N)
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(sieve1(X)) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
POL(0) = 0
POL(A__FILTER3(x1, x2, x3)) = 2·x1
POL(A__NATS1(x1)) = 2·x1
POL(A__SIEVE1(x1)) = x1
POL(MARK1(x1)) = x1
POL(a__filter3(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(a__nats1(x1)) = 1 + 2·x1
POL(a__sieve1(x1)) = 2·x1
POL(a__zprimes) = 2
POL(cons2(x1, x2)) = 2·x1
POL(filter3(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(mark1(x1)) = x1
POL(nats1(x1)) = 1 + 2·x1
POL(s1(x1)) = 2·x1
POL(sieve1(x1)) = 2·x1
POL(zprimes) = 2
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__zprimes -> zprimes
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
a__nats1(X) -> nats1(X)
mark1(0) -> 0
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
a__sieve1(X) -> sieve1(X)
mark1(nats1(X)) -> a__nats1(mark1(X))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
mark1(zprimes) -> a__zprimes
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__NATS1(N) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(sieve1(X)) -> MARK1(X)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(sieve1(X)) -> MARK1(X)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__SIEVE1(cons2(s1(N), Y)) -> MARK1(N)
A__FILTER3(cons2(X, Y), s1(N), M) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)
Used ordering: Polynomial interpretation [21]:
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
MARK1(sieve1(X)) -> MARK1(X)
POL(0) = 0
POL(A__FILTER3(x1, x2, x3)) = 2·x1
POL(A__SIEVE1(x1)) = x1
POL(MARK1(x1)) = x1
POL(a__filter3(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(a__nats1(x1)) = 1 + 2·x1
POL(a__sieve1(x1)) = 2·x1
POL(a__zprimes) = 2
POL(cons2(x1, x2)) = 1 + 2·x1
POL(filter3(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(mark1(x1)) = x1
POL(nats1(x1)) = 1 + 2·x1
POL(s1(x1)) = 2·x1
POL(sieve1(x1)) = 2·x1
POL(zprimes) = 2
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__zprimes -> zprimes
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
a__nats1(X) -> nats1(X)
mark1(0) -> 0
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
a__sieve1(X) -> sieve1(X)
mark1(nats1(X)) -> a__nats1(mark1(X))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
mark1(zprimes) -> a__zprimes
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK1(filter3(X1, X2, X3)) -> A__FILTER3(mark1(X1), mark1(X2), mark1(X3))
MARK1(s1(X)) -> MARK1(X)
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(sieve1(X)) -> A__SIEVE1(mark1(X))
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
MARK1(sieve1(X)) -> MARK1(X)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
MARK1(s1(X)) -> MARK1(X)
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
MARK1(sieve1(X)) -> MARK1(X)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK1(sieve1(X)) -> MARK1(X)
Used ordering: Polynomial interpretation [21]:
MARK1(s1(X)) -> MARK1(X)
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
POL(MARK1(x1)) = 2·x1
POL(filter3(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(s1(x1)) = 2·x1
POL(sieve1(x1)) = 2 + 2·x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
MARK1(s1(X)) -> MARK1(X)
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK1(filter3(X1, X2, X3)) -> MARK1(X1)
MARK1(filter3(X1, X2, X3)) -> MARK1(X3)
MARK1(filter3(X1, X2, X3)) -> MARK1(X2)
Used ordering: Polynomial interpretation [21]:
MARK1(s1(X)) -> MARK1(X)
POL(MARK1(x1)) = 2·x1
POL(filter3(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3
POL(s1(x1)) = 2·x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
MARK1(s1(X)) -> MARK1(X)
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK1(s1(X)) -> MARK1(X)
POL(MARK1(x1)) = 2·x1
POL(s1(x1)) = 1 + 2·x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
a__filter3(cons2(X, Y), 0, M) -> cons2(0, filter3(Y, M, M))
a__filter3(cons2(X, Y), s1(N), M) -> cons2(mark1(X), filter3(Y, N, M))
a__sieve1(cons2(0, Y)) -> cons2(0, sieve1(Y))
a__sieve1(cons2(s1(N), Y)) -> cons2(s1(mark1(N)), sieve1(filter3(Y, N, N)))
a__nats1(N) -> cons2(mark1(N), nats1(s1(N)))
a__zprimes -> a__sieve1(a__nats1(s1(s1(0))))
mark1(filter3(X1, X2, X3)) -> a__filter3(mark1(X1), mark1(X2), mark1(X3))
mark1(sieve1(X)) -> a__sieve1(mark1(X))
mark1(nats1(X)) -> a__nats1(mark1(X))
mark1(zprimes) -> a__zprimes
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(mark1(X))
a__filter3(X1, X2, X3) -> filter3(X1, X2, X3)
a__sieve1(X) -> sieve1(X)
a__nats1(X) -> nats1(X)
a__zprimes -> zprimes