YES Problem: 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() Proof: DP Processor: DPs: 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__nats#(s(s(0()))) a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(nats(X)) -> mark#(X) mark#(nats(X)) -> a__nats#(mark(X)) mark#(zprimes()) -> a__zprimes#() mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: 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() TDG Processor: DPs: 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__nats#(s(s(0()))) a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(nats(X)) -> mark#(X) mark#(nats(X)) -> a__nats#(mark(X)) mark#(zprimes()) -> a__zprimes#() mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: 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() graph: a__zprimes#() -> a__nats#(s(s(0()))) -> a__nats#(N) -> mark#(N) a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) -> a__sieve#(cons(s(N),Y)) -> mark#(N) a__nats#(N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__nats#(N) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__nats#(N) -> mark#(N) -> mark#(zprimes()) -> a__zprimes#() a__nats#(N) -> mark#(N) -> mark#(nats(X)) -> a__nats#(mark(X)) a__nats#(N) -> mark#(N) -> mark#(nats(X)) -> mark#(X) a__nats#(N) -> mark#(N) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__nats#(N) -> mark#(N) -> mark#(sieve(X)) -> mark#(X) a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X1) a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X2) a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X3) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(zprimes()) -> a__zprimes#() a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(nats(X)) -> a__nats#(mark(X)) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(nats(X)) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(sieve(X)) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X1) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X2) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(zprimes()) -> a__zprimes#() -> a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) mark#(zprimes()) -> a__zprimes#() -> a__zprimes#() -> a__nats#(s(s(0()))) mark#(nats(X)) -> a__nats#(mark(X)) -> a__nats#(N) -> mark#(N) mark#(nats(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(nats(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(nats(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#() mark#(nats(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(nats(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X) mark#(nats(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(nats(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(nats(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#(sieve(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#() mark#(sieve(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#() mark#(s(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(zprimes()) -> a__zprimes#() mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(nats(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(zprimes()) -> a__zprimes#() mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(nats(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X2) -> 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#(X1) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(zprimes()) -> a__zprimes#() mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(nats(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X1) -> 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#(X1) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) -> a__filter#(cons(X,Y),s(N),M) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zprimes()) -> a__zprimes#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X3) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#() a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X)) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(nats(X)) -> mark#(X) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3) Matrix Interpretation Processor: dim=1 interpretation: [a__zprimes#] = 5/2, [a__nats#](x0) = x0 + 1/2, [a__sieve#](x0) = 1/2x0, [mark#](x0) = x0, [a__filter#](x0, x1, x2) = x0 + 1/2, [zprimes] = 7/2, [a__zprimes] = 7/2, [nats](x0) = x0 + 3/2, [a__nats](x0) = x0 + 3/2, [sieve](x0) = x0 + 1/2, [a__sieve](x0) = x0 + 1/2, [mark](x0) = x0, [s](x0) = 2x0 + 1/2, [filter](x0, x1, x2) = 2x0 + 3x1 + x2 + 1, [a__filter](x0, x1, x2) = 2x0 + 3x1 + x2 + 1, [0] = 0, [cons](x0, x1) = x0 + 3/2 orientation: a__filter#(cons(X,Y),s(N),M) = X + 2 >= X = mark#(X) a__sieve#(cons(s(N),Y)) = N + 1 >= N = mark#(N) a__nats#(N) = N + 1/2 >= N = mark#(N) a__zprimes#() = 5/2 >= 2 = a__nats#(s(s(0()))) a__zprimes#() = 5/2 >= 3/2 = a__sieve#(a__nats(s(s(0())))) mark#(filter(X1,X2,X3)) = 2X1 + 3X2 + X3 + 1 >= X3 = mark#(X3) mark#(filter(X1,X2,X3)) = 2X1 + 3X2 + X3 + 1 >= X2 = mark#(X2) mark#(filter(X1,X2,X3)) = 2X1 + 3X2 + X3 + 1 >= X1 = mark#(X1) mark#(filter(X1,X2,X3)) = 2X1 + 3X2 + X3 + 1 >= X1 + 1/2 = a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) = X + 1/2 >= X = mark#(X) mark#(sieve(X)) = X + 1/2 >= 1/2X = a__sieve#(mark(X)) mark#(nats(X)) = X + 3/2 >= X = mark#(X) mark#(nats(X)) = X + 3/2 >= X + 1/2 = a__nats#(mark(X)) mark#(zprimes()) = 7/2 >= 5/2 = a__zprimes#() mark#(cons(X1,X2)) = X1 + 3/2 >= X1 = mark#(X1) mark#(s(X)) = 2X + 1/2 >= X = mark#(X) a__filter(cons(X,Y),0(),M) = M + 2X + 4 >= 3/2 = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = M + 6N + 2X + 11/2 >= X + 3/2 = cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) = 2 >= 3/2 = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = 2N + 5/2 >= 2N + 2 = cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) = N + 3/2 >= N + 3/2 = cons(mark(N),nats(s(N))) a__zprimes() = 7/2 >= 7/2 = a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) = 2X1 + 3X2 + X3 + 1 >= 2X1 + 3X2 + X3 + 1 = a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) = X + 1/2 >= X + 1/2 = a__sieve(mark(X)) mark(nats(X)) = X + 3/2 >= X + 3/2 = a__nats(mark(X)) mark(zprimes()) = 7/2 >= 7/2 = a__zprimes() mark(cons(X1,X2)) = X1 + 3/2 >= X1 + 3/2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 2X + 1/2 >= 2X + 1/2 = s(mark(X)) a__filter(X1,X2,X3) = 2X1 + 3X2 + X3 + 1 >= 2X1 + 3X2 + X3 + 1 = filter(X1,X2,X3) a__sieve(X) = X + 1/2 >= X + 1/2 = sieve(X) a__nats(X) = X + 3/2 >= X + 3/2 = nats(X) a__zprimes() = 7/2 >= 7/2 = zprimes() problem: DPs: TRS: 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() Qed