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) Arctic Interpretation Processor: dimension: 1 interpretation: [a__zprimes#] = 7, [a__nats#](x0) = 2x0 + 0, [a__sieve#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__filter#](x0, x1, x2) = x0 + x1 + x2 + 0, [zprimes] = 7, [a__zprimes] = 7, [nats](x0) = 2x0 + 2, [a__nats](x0) = 2x0 + 2, [sieve](x0) = x0 + 4, [a__sieve](x0) = x0 + 4, [mark](x0) = x0 + 0, [s](x0) = x0 + 0, [filter](x0, x1, x2) = x0 + x1 + x2, [a__filter](x0, x1, x2) = x0 + x1 + x2 + 0, [0] = 5, [cons](x0, x1) = x0 + 0 orientation: a__filter#(cons(X,Y),s(N),M) = M + N + X + 0 >= X + 0 = mark#(X) a__sieve#(cons(s(N),Y)) = N + 0 >= N + 0 = mark#(N) a__nats#(N) = 2N + 0 >= N + 0 = mark#(N) a__zprimes#() = 7 >= 7 = a__nats#(s(s(0()))) a__zprimes#() = 7 >= 7 = a__sieve#(a__nats(s(s(0())))) mark#(filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X3 + 0 = mark#(X3) mark#(filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X2 + 0 = mark#(X2) mark#(filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) = X + 4 >= X + 0 = mark#(X) mark#(sieve(X)) = X + 4 >= X + 0 = a__sieve#(mark(X)) mark#(nats(X)) = 2X + 2 >= X + 0 = mark#(X) mark#(nats(X)) = 2X + 2 >= 2X + 2 = a__nats#(mark(X)) mark#(zprimes()) = 7 >= 7 = a__zprimes#() mark#(cons(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__filter(cons(X,Y),0(),M) = M + X + 5 >= 5 = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = M + N + X + 0 >= X + 0 = cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) = 5 >= 5 = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = N + 4 >= N + 0 = cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) = 2N + 2 >= N + 0 = cons(mark(N),nats(s(N))) a__zprimes() = 7 >= 7 = a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) = X + 4 >= X + 4 = a__sieve(mark(X)) mark(nats(X)) = 2X + 2 >= 2X + 2 = a__nats(mark(X)) mark(zprimes()) = 7 >= 7 = a__zprimes() mark(cons(X1,X2)) = X1 + 0 >= X1 + 0 = cons(mark(X1),X2) mark(0()) = 5 >= 5 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) a__filter(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 = filter(X1,X2,X3) a__sieve(X) = X + 4 >= X + 4 = sieve(X) a__nats(X) = 2X + 2 >= 2X + 2 = nats(X) a__zprimes() = 7 >= 7 = zprimes() problem: 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)) -> 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() Arctic Interpretation Processor: dimension: 1 interpretation: [a__zprimes#] = 7, [a__nats#](x0) = 6x0, [a__sieve#](x0) = x0 + 0, [mark#](x0) = x0, [a__filter#](x0, x1, x2) = 1x0 + x1, [zprimes] = 7, [a__zprimes] = 7, [nats](x0) = 7x0 + 0, [a__nats](x0) = 7x0 + 0, [sieve](x0) = x0 + 0, [a__sieve](x0) = x0 + 0, [mark](x0) = x0, [s](x0) = x0 + 0, [filter](x0, x1, x2) = 1x0 + x1 + x2, [a__filter](x0, x1, x2) = 1x0 + x1 + x2, [0] = 0, [cons](x0, x1) = x0 orientation: a__filter#(cons(X,Y),s(N),M) = N + 1X + 0 >= X = mark#(X) a__sieve#(cons(s(N),Y)) = N + 0 >= N = mark#(N) a__nats#(N) = 6N >= N = mark#(N) a__zprimes#() = 7 >= 6 = a__nats#(s(s(0()))) a__zprimes#() = 7 >= 7 = a__sieve#(a__nats(s(s(0())))) mark#(filter(X1,X2,X3)) = 1X1 + X2 + X3 >= X3 = mark#(X3) mark#(filter(X1,X2,X3)) = 1X1 + X2 + X3 >= X2 = mark#(X2) mark#(filter(X1,X2,X3)) = 1X1 + X2 + X3 >= X1 = mark#(X1) mark#(filter(X1,X2,X3)) = 1X1 + X2 + X3 >= 1X1 + X2 = a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) = X + 0 >= X = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = a__sieve#(mark(X)) mark#(nats(X)) = 7X + 0 >= 6X = a__nats#(mark(X)) mark#(zprimes()) = 7 >= 7 = a__zprimes#() mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) mark#(s(X)) = X + 0 >= X = mark#(X) a__filter(cons(X,Y),0(),M) = M + 1X + 0 >= 0 = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = M + N + 1X + 0 >= X = cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) = 0 >= 0 = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = N + 0 >= N + 0 = cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) = 7N + 0 >= N = cons(mark(N),nats(s(N))) a__zprimes() = 7 >= 7 = a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) = 1X1 + X2 + X3 >= 1X1 + X2 + X3 = a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(nats(X)) = 7X + 0 >= 7X + 0 = a__nats(mark(X)) mark(zprimes()) = 7 >= 7 = a__zprimes() mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) a__filter(X1,X2,X3) = 1X1 + X2 + X3 >= 1X1 + X2 + X3 = filter(X1,X2,X3) a__sieve(X) = X + 0 >= X + 0 = sieve(X) a__nats(X) = 7X + 0 >= 7X + 0 = nats(X) a__zprimes() = 7 >= 7 = zprimes() problem: DPs: a__sieve#(cons(s(N),Y)) -> mark#(N) 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)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(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() SCC Processor: #sccs: 1 #rules: 9 #arcs: 117/100 DPs: a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) a__sieve#(cons(s(N),Y)) -> mark#(N) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(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() Arctic Interpretation Processor: dimension: 1 interpretation: [a__zprimes#] = 0, [a__sieve#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [zprimes] = 0, [a__zprimes] = 0, [nats](x0) = x0, [a__nats](x0) = x0 + 0, [sieve](x0) = x0, [a__sieve](x0) = x0 + 0, [mark](x0) = x0 + 0, [s](x0) = x0 + 0, [filter](x0, x1, x2) = 4x0 + 4x1 + x2 + 4, [a__filter](x0, x1, x2) = 4x0 + 4x1 + x2 + 4, [0] = 0, [cons](x0, x1) = x0 + 0 orientation: a__zprimes#() = 0 >= 0 = a__sieve#(a__nats(s(s(0())))) a__sieve#(cons(s(N),Y)) = N + 0 >= N + 0 = mark#(N) mark#(filter(X1,X2,X3)) = 4X1 + 4X2 + X3 + 4 >= X3 + 0 = mark#(X3) mark#(filter(X1,X2,X3)) = 4X1 + 4X2 + X3 + 4 >= X2 + 0 = mark#(X2) mark#(sieve(X)) = X + 0 >= X + 0 = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = a__sieve#(mark(X)) mark#(zprimes()) = 0 >= 0 = a__zprimes#() mark#(cons(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__filter(cons(X,Y),0(),M) = M + 4X + 4 >= 0 = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = M + 4N + 4X + 4 >= X + 0 = cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) = 0 >= 0 = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = N + 0 >= N + 0 = cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) = N + 0 >= N + 0 = cons(mark(N),nats(s(N))) a__zprimes() = 0 >= 0 = a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) = 4X1 + 4X2 + X3 + 4 >= 4X1 + 4X2 + X3 + 4 = a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(nats(X)) = X + 0 >= X + 0 = a__nats(mark(X)) mark(zprimes()) = 0 >= 0 = a__zprimes() mark(cons(X1,X2)) = X1 + 0 >= X1 + 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) a__filter(X1,X2,X3) = 4X1 + 4X2 + X3 + 4 >= 4X1 + 4X2 + X3 + 4 = filter(X1,X2,X3) a__sieve(X) = X + 0 >= X = sieve(X) a__nats(X) = X + 0 >= X = nats(X) a__zprimes() = 0 >= 0 = zprimes() problem: DPs: a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) a__sieve#(cons(s(N),Y)) -> mark#(N) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(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() Arctic Interpretation Processor: dimension: 1 interpretation: [a__zprimes#] = 0, [a__sieve#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [zprimes] = 2, [a__zprimes] = 2, [nats](x0) = x0 + 0, [a__nats](x0) = x0 + 0, [sieve](x0) = x0 + 0, [a__sieve](x0) = x0 + 0, [mark](x0) = x0, [s](x0) = x0, [filter](x0, x1, x2) = x0 + x1 + 4x2, [a__filter](x0, x1, x2) = x0 + x1 + 4x2, [0] = 0, [cons](x0, x1) = x0 orientation: a__zprimes#() = 0 >= 0 = a__sieve#(a__nats(s(s(0())))) a__sieve#(cons(s(N),Y)) = N + 0 >= N + 0 = mark#(N) mark#(filter(X1,X2,X3)) = X1 + X2 + 4X3 + 0 >= X3 + 0 = mark#(X3) mark#(sieve(X)) = X + 0 >= X + 0 = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = a__sieve#(mark(X)) mark#(zprimes()) = 2 >= 0 = a__zprimes#() mark#(cons(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__filter(cons(X,Y),0(),M) = 4M + X + 0 >= 0 = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = 4M + N + X >= X = cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) = 0 >= 0 = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = N + 0 >= N = cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) = N + 0 >= N = cons(mark(N),nats(s(N))) a__zprimes() = 2 >= 0 = a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) = X1 + X2 + 4X3 >= X1 + X2 + 4X3 = a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(nats(X)) = X + 0 >= X + 0 = a__nats(mark(X)) mark(zprimes()) = 2 >= 2 = a__zprimes() mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) a__filter(X1,X2,X3) = X1 + X2 + 4X3 >= X1 + X2 + 4X3 = filter(X1,X2,X3) a__sieve(X) = X + 0 >= X + 0 = sieve(X) a__nats(X) = X + 0 >= X + 0 = nats(X) a__zprimes() = 2 >= 2 = zprimes() problem: DPs: a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) a__sieve#(cons(s(N),Y)) -> mark#(N) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) 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() SCC Processor: #sccs: 1 #rules: 6 #arcs: 45/49 DPs: a__sieve#(cons(s(N),Y)) -> mark#(N) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) 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() Arctic Interpretation Processor: dimension: 1 interpretation: [a__sieve#](x0) = 1x0 + 0, [mark#](x0) = x0 + 1, [zprimes] = 3, [a__zprimes] = 3, [nats](x0) = x0 + 0, [a__nats](x0) = x0 + 0, [sieve](x0) = 1x0 + 2, [a__sieve](x0) = 1x0 + 2, [mark](x0) = x0, [s](x0) = x0, [filter](x0, x1, x2) = x0 + x1 + x2, [a__filter](x0, x1, x2) = x0 + x1 + x2, [0] = 2, [cons](x0, x1) = x0 + x1 + 0 orientation: a__sieve#(cons(s(N),Y)) = 1N + 1Y + 1 >= N + 1 = mark#(N) mark#(filter(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X3 + 1 = mark#(X3) mark#(sieve(X)) = 1X + 2 >= X + 1 = mark#(X) mark#(sieve(X)) = 1X + 2 >= 1X + 0 = a__sieve#(mark(X)) mark#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) a__filter(cons(X,Y),0(),M) = M + X + Y + 2 >= M + Y + 2 = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = M + N + X + Y + 0 >= M + N + X + Y + 0 = cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) = 1Y + 3 >= 1Y + 2 = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = 1N + 1Y + 2 >= 1N + 1Y + 2 = cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) = N + 0 >= N + 0 = cons(mark(N),nats(s(N))) a__zprimes() = 3 >= 3 = a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) = 1X + 2 >= 1X + 2 = a__sieve(mark(X)) mark(nats(X)) = X + 0 >= X + 0 = a__nats(mark(X)) mark(zprimes()) = 3 >= 3 = a__zprimes() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 2 >= 2 = 0() mark(s(X)) = X >= X = s(mark(X)) a__filter(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = filter(X1,X2,X3) a__sieve(X) = 1X + 2 >= 1X + 2 = sieve(X) a__nats(X) = X + 0 >= X + 0 = nats(X) a__zprimes() = 3 >= 3 = zprimes() problem: DPs: a__sieve#(cons(s(N),Y)) -> mark#(N) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(sieve(X)) -> a__sieve#(mark(X)) 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() Arctic Interpretation Processor: dimension: 1 interpretation: [a__sieve#](x0) = x0, [mark#](x0) = x0, [zprimes] = 7, [a__zprimes] = 7, [nats](x0) = x0, [a__nats](x0) = x0, [sieve](x0) = 1x0 + 0, [a__sieve](x0) = 1x0 + 0, [mark](x0) = x0, [s](x0) = 3x0 + 0, [filter](x0, x1, x2) = 1x0 + x1 + x2, [a__filter](x0, x1, x2) = 1x0 + x1 + x2, [0] = 0, [cons](x0, x1) = x0 orientation: a__sieve#(cons(s(N),Y)) = 3N + 0 >= N = mark#(N) mark#(filter(X1,X2,X3)) = 1X1 + X2 + X3 >= X3 = mark#(X3) mark#(sieve(X)) = 1X + 0 >= X = a__sieve#(mark(X)) mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) mark#(s(X)) = 3X + 0 >= X = mark#(X) a__filter(cons(X,Y),0(),M) = M + 1X + 0 >= 0 = cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) = M + 3N + 1X + 0 >= X = cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) = 1 >= 0 = cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) = 4N + 1 >= 3N + 0 = cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) = N >= N = cons(mark(N),nats(s(N))) a__zprimes() = 7 >= 7 = a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) = 1X1 + X2 + X3 >= 1X1 + X2 + X3 = a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) = 1X + 0 >= 1X + 0 = a__sieve(mark(X)) mark(nats(X)) = X >= X = a__nats(mark(X)) mark(zprimes()) = 7 >= 7 = a__zprimes() mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 3X + 0 >= 3X + 0 = s(mark(X)) a__filter(X1,X2,X3) = 1X1 + X2 + X3 >= 1X1 + X2 + X3 = filter(X1,X2,X3) a__sieve(X) = 1X + 0 >= 1X + 0 = sieve(X) a__nats(X) = X >= X = nats(X) a__zprimes() = 7 >= 7 = zprimes() problem: DPs: mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(cons(X1,X2)) -> mark#(X1) 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() Subterm Criterion Processor: simple projection: pi(mark#) = 0 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