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
↳ QTRS
↳ DependencyPairsProof
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
MARK(zprimes) → A__ZPRIMES
MARK(s(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X3)
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(nats(X)) → A__NATS(mark(X))
MARK(filter(X1, X2, X3)) → MARK(X2)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(nats(X)) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
A__NATS(N) → MARK(N)
A__ZPRIMES → A__NATS(s(s(0)))
A__ZPRIMES → A__SIEVE(a__nats(s(s(0))))
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
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
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
MARK(zprimes) → A__ZPRIMES
MARK(s(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X3)
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(nats(X)) → A__NATS(mark(X))
MARK(filter(X1, X2, X3)) → MARK(X2)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(nats(X)) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
A__NATS(N) → MARK(N)
A__ZPRIMES → A__NATS(s(s(0)))
A__ZPRIMES → A__SIEVE(a__nats(s(s(0))))
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
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
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__ZPRIMES → A__NATS(s(s(0)))
A__ZPRIMES → A__SIEVE(a__nats(s(s(0))))
Used ordering: Polynomial interpretation [25]:
MARK(zprimes) → A__ZPRIMES
MARK(s(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X3)
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(nats(X)) → A__NATS(mark(X))
MARK(filter(X1, X2, X3)) → MARK(X2)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(nats(X)) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
A__NATS(N) → MARK(N)
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
POL(0) = 0
POL(A__FILTER(x1, x2, x3)) = x1
POL(A__NATS(x1)) = x1
POL(A__SIEVE(x1)) = x1
POL(A__ZPRIMES) = 1
POL(MARK(x1)) = x1
POL(a__filter(x1, x2, x3)) = x1 + x2 + x3
POL(a__nats(x1)) = x1
POL(a__sieve(x1)) = x1
POL(a__zprimes) = 1
POL(cons(x1, x2)) = x1
POL(filter(x1, x2, x3)) = x1 + x2 + x3
POL(mark(x1)) = x1
POL(nats(x1)) = x1
POL(s(x1)) = x1
POL(sieve(x1)) = x1
POL(zprimes) = 1
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__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(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
mark(zprimes) → a__zprimes
mark(nats(X)) → a__nats(mark(X))
mark(sieve(X)) → a__sieve(mark(X))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__zprimes → zprimes
a__nats(X) → nats(X)
a__sieve(X) → sieve(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK(zprimes) → A__ZPRIMES
MARK(s(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X3)
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(nats(X)) → A__NATS(mark(X))
MARK(filter(X1, X2, X3)) → MARK(X2)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(nats(X)) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
A__NATS(N) → MARK(N)
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
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
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
MARK(filter(X1, X2, X3)) → MARK(X3)
MARK(s(X)) → MARK(X)
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(nats(X)) → A__NATS(mark(X))
MARK(filter(X1, X2, X3)) → MARK(X2)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(nats(X)) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
A__NATS(N) → MARK(N)
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
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
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(nats(X)) → MARK(X)
A__NATS(N) → MARK(N)
Used ordering: Polynomial interpretation [25]:
MARK(filter(X1, X2, X3)) → MARK(X3)
MARK(s(X)) → MARK(X)
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(nats(X)) → A__NATS(mark(X))
MARK(filter(X1, X2, X3)) → MARK(X2)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
POL(0) = 0
POL(A__FILTER(x1, x2, x3)) = x1
POL(A__NATS(x1)) = 1 + x1
POL(A__SIEVE(x1)) = x1
POL(MARK(x1)) = x1
POL(a__filter(x1, x2, x3)) = x1 + x2 + x3
POL(a__nats(x1)) = 1 + x1
POL(a__sieve(x1)) = x1
POL(a__zprimes) = 1
POL(cons(x1, x2)) = x1
POL(filter(x1, x2, x3)) = x1 + x2 + x3
POL(mark(x1)) = x1
POL(nats(x1)) = 1 + x1
POL(s(x1)) = x1
POL(sieve(x1)) = x1
POL(zprimes) = 1
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__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(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
mark(zprimes) → a__zprimes
mark(nats(X)) → a__nats(mark(X))
mark(sieve(X)) → a__sieve(mark(X))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__zprimes → zprimes
a__nats(X) → nats(X)
a__sieve(X) → sieve(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(nats(X)) → A__NATS(mark(X))
MARK(filter(X1, X2, X3)) → MARK(X2)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X3)
MARK(sieve(X)) → A__SIEVE(mark(X))
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
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
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDPOrderProof
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(filter(X1, X2, X3)) → MARK(X2)
MARK(s(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X3)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
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
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(filter(X1, X2, X3)) → MARK(X2)
MARK(filter(X1, X2, X3)) → MARK(X3)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(filter(X1, X2, X3)) → MARK(X1)
Used ordering: Polynomial interpretation with max and min functions [25]:
MARK(s(X)) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
MARK(sieve(X)) → MARK(X)
POL(0) = 0
POL(A__FILTER(x1, x2, x3)) = x1
POL(A__SIEVE(x1)) = x1
POL(MARK(x1)) = x1
POL(a__filter(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__nats(x1)) = 1 + x1
POL(a__sieve(x1)) = x1
POL(a__zprimes) = 1
POL(cons(x1, x2)) = 1 + x1
POL(filter(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(mark(x1)) = x1
POL(nats(x1)) = 1 + x1
POL(s(x1)) = x1
POL(sieve(x1)) = x1
POL(zprimes) = 1
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__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(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
mark(zprimes) → a__zprimes
mark(nats(X)) → a__nats(mark(X))
mark(sieve(X)) → a__sieve(mark(X))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__zprimes → zprimes
a__nats(X) → nats(X)
a__sieve(X) → sieve(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDPOrderProof
MARK(s(X)) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
MARK(sieve(X)) → MARK(X)
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
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDPOrderProof
MARK(s(X)) → MARK(X)
MARK(sieve(X)) → MARK(X)
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
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDPOrderProof
MARK(s(X)) → MARK(X)
MARK(sieve(X)) → MARK(X)
From the DPs we obtained the following set of size-change graphs:
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(filter(X1, X2, X3)) → A__FILTER(mark(X1), mark(X2), mark(X3))
MARK(filter(X1, X2, X3)) → MARK(X2)
MARK(filter(X1, X2, X3)) → MARK(X3)
A__SIEVE(cons(s(N), Y)) → MARK(N)
MARK(sieve(X)) → MARK(X)
MARK(filter(X1, X2, X3)) → MARK(X1)
Used ordering: Polynomial interpretation [25]:
MARK(s(X)) → MARK(X)
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
POL(0) = 0
POL(A__FILTER(x1, x2, x3)) = x1
POL(A__SIEVE(x1)) = 1 + x1
POL(MARK(x1)) = x1
POL(a__filter(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__nats(x1)) = x1
POL(a__sieve(x1)) = 1 + x1
POL(a__zprimes) = 1
POL(cons(x1, x2)) = x1
POL(filter(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(mark(x1)) = x1
POL(nats(x1)) = x1
POL(s(x1)) = x1
POL(sieve(x1)) = 1 + x1
POL(zprimes) = 1
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__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(cons(s(N), Y)) → cons(s(mark(N)), sieve(filter(Y, N, N)))
mark(zprimes) → a__zprimes
mark(nats(X)) → a__nats(mark(X))
mark(sieve(X)) → a__sieve(mark(X))
mark(filter(X1, X2, X3)) → a__filter(mark(X1), mark(X2), mark(X3))
a__filter(X1, X2, X3) → filter(X1, X2, X3)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__zprimes → zprimes
a__nats(X) → nats(X)
a__sieve(X) → sieve(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDPOrderProof
↳ QDP
A__FILTER(cons(X, Y), s(N), M) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(sieve(X)) → A__SIEVE(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
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