YES Problem: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Proof: DP Processor: DPs: filter#(cons(X,Y),0(),M) -> activate#(Y) filter#(cons(X,Y),s(N),M) -> activate#(Y) sieve#(cons(0(),Y)) -> activate#(Y) sieve#(cons(s(N),Y)) -> activate#(Y) zprimes#() -> s#(0()) zprimes#() -> s#(s(0())) zprimes#() -> nats#(s(s(0()))) zprimes#() -> sieve#(nats(s(s(0())))) activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__nats(X)) -> activate#(X) activate#(n__nats(X)) -> nats#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X TDG Processor: DPs: filter#(cons(X,Y),0(),M) -> activate#(Y) filter#(cons(X,Y),s(N),M) -> activate#(Y) sieve#(cons(0(),Y)) -> activate#(Y) sieve#(cons(s(N),Y)) -> activate#(Y) zprimes#() -> s#(0()) zprimes#() -> s#(s(0())) zprimes#() -> nats#(s(s(0()))) zprimes#() -> sieve#(nats(s(s(0())))) activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__nats(X)) -> activate#(X) activate#(n__nats(X)) -> nats#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X graph: zprimes#() -> sieve#(nats(s(s(0())))) -> sieve#(cons(s(N),Y)) -> activate#(Y) zprimes#() -> sieve#(nats(s(s(0())))) -> sieve#(cons(0(),Y)) -> activate#(Y) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__s(X)) -> s#(activate(X)) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__s(X)) -> activate#(X) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__nats(X)) -> nats#(activate(X)) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__nats(X)) -> activate#(X) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__sieve(X)) -> sieve#(activate(X)) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__sieve(X)) -> activate#(X) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) sieve#(cons(s(N),Y)) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__s(X)) -> s#(activate(X)) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__s(X)) -> activate#(X) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__nats(X)) -> nats#(activate(X)) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__nats(X)) -> activate#(X) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__sieve(X)) -> sieve#(activate(X)) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__sieve(X)) -> activate#(X) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) sieve#(cons(0(),Y)) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__nats(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__nats(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__nats(X)) -> activate#(X) -> activate#(n__nats(X)) -> nats#(activate(X)) activate#(n__nats(X)) -> activate#(X) -> activate#(n__nats(X)) -> activate#(X) activate#(n__nats(X)) -> activate#(X) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__nats(X)) -> activate#(X) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__nats(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) activate#(n__nats(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__nats(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__nats(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__nats(X)) -> nats#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__nats(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) activate#(n__s(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__sieve(X)) -> sieve#(activate(X)) -> sieve#(cons(s(N),Y)) -> activate#(Y) activate#(n__sieve(X)) -> sieve#(activate(X)) -> sieve#(cons(0(),Y)) -> activate#(Y) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__nats(X)) -> nats#(activate(X)) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__nats(X)) -> activate#(X) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__s(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__nats(X)) -> nats#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__nats(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X3) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__nats(X)) -> nats#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__nats(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X2) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__nats(X)) -> nats#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__nats(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X1) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) -> filter#(cons(X,Y),s(N),M) -> activate#(Y) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) -> filter#(cons(X,Y),0(),M) -> activate#(Y) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__s(X)) -> s#(activate(X)) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__nats(X)) -> nats#(activate(X)) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__nats(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__sieve(X)) -> sieve#(activate(X)) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__sieve(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) filter#(cons(X,Y),s(N),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__s(X)) -> s#(activate(X)) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__nats(X)) -> nats#(activate(X)) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__nats(X)) -> activate#(X) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__sieve(X)) -> sieve#(activate(X)) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__sieve(X)) -> activate#(X) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X1) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X2) filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> activate#(X3) SCC Processor: #sccs: 1 #rules: 12 #arcs: 106/324 DPs: sieve#(cons(0(),Y)) -> activate#(Y) activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) -> activate#(Y) activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> sieve#(activate(X)) sieve#(cons(s(N),Y)) -> activate#(Y) activate#(n__nats(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [sieve#](x0) = 2x0 + 0, [activate#](x0) = 1x0, [filter#](x0, x1, x2) = 1x0 + 1x1 + 0, [zprimes] = 1, [n__nats](x0) = x0 + 0, [n__s](x0) = x0 + 0, [nats](x0) = x0 + 0, [n__sieve](x0) = 1x0 + 1, [sieve](x0) = 1x0 + 1, [s](x0) = x0 + 0, [n__filter](x0, x1, x2) = x0 + x1 + x2 + 0, [activate](x0) = x0 + 0, [filter](x0, x1, x2) = x0 + x1 + x2 + 0, [0] = 0, [cons](x0, x1) = x0 + x1 + 0 orientation: sieve#(cons(0(),Y)) = 2Y + 2 >= 1Y = activate#(Y) activate#(n__filter(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 1 >= 1X3 = activate#(X3) activate#(n__filter(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 1 >= 1X2 = activate#(X2) activate#(n__filter(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 1 >= 1X1 = activate#(X1) activate#(n__filter(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 1 >= 1X1 + 1X2 + 1 = filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) = 1X + 1Y + 1 >= 1Y = activate#(Y) activate#(n__sieve(X)) = 2X + 2 >= 1X = activate#(X) activate#(n__sieve(X)) = 2X + 2 >= 2X + 2 = sieve#(activate(X)) sieve#(cons(s(N),Y)) = 2N + 2Y + 2 >= 1Y = activate#(Y) activate#(n__nats(X)) = 1X + 1 >= 1X = activate#(X) activate#(n__s(X)) = 1X + 1 >= 1X = activate#(X) filter#(cons(X,Y),s(N),M) = 1N + 1X + 1Y + 1 >= 1Y = activate#(Y) filter(cons(X,Y),0(),M) = M + X + Y + 0 >= M + Y + 0 = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = M + N + X + Y + 0 >= M + N + X + Y + 0 = cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) = 1Y + 1 >= 1Y + 1 = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = 1N + 1Y + 1 >= 1N + 1Y + 1 = cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) = N + 0 >= N + 0 = cons(N,n__nats(n__s(N))) zprimes() = 1 >= 1 = sieve(nats(s(s(0())))) filter(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = n__filter(X1,X2,X3) sieve(X) = 1X + 1 >= 1X + 1 = n__sieve(X) nats(X) = X + 0 >= X + 0 = n__nats(X) s(X) = X + 0 >= X + 0 = n__s(X) activate(n__filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) = 1X + 1 >= 1X + 1 = sieve(activate(X)) activate(n__nats(X)) = X + 0 >= X + 0 = nats(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(X) = X + 0 >= X = X problem: DPs: activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) -> activate#(Y) activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__nats(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 8 #arcs: 84/81 DPs: activate#(n__nats(X)) -> activate#(X) activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) -> activate#(Y) activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = x0 + 0, [filter#](x0, x1, x2) = x0 + x1 + x2 + 0, [zprimes] = 9, [n__nats](x0) = 4x0 + 4, [n__s](x0) = x0 + 0, [nats](x0) = 4x0 + 4, [n__sieve](x0) = x0, [sieve](x0) = x0 + 0, [s](x0) = x0 + 0, [n__filter](x0, x1, x2) = x0 + x1 + x2 + 0, [activate](x0) = x0 + 0, [filter](x0, x1, x2) = x0 + x1 + x2 + 0, [0] = 0, [cons](x0, x1) = x0 + x1 + 0 orientation: activate#(n__nats(X)) = 4X + 4 >= X + 0 = activate#(X) activate#(n__filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X3 + 0 = activate#(X3) activate#(n__filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X2 + 0 = activate#(X2) activate#(n__filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = activate#(X1) activate#(n__filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) = M + X + Y + 0 >= Y + 0 = activate#(Y) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) filter#(cons(X,Y),s(N),M) = M + N + X + Y + 0 >= Y + 0 = activate#(Y) filter(cons(X,Y),0(),M) = M + X + Y + 0 >= M + Y + 0 = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = M + N + X + Y + 0 >= M + N + X + Y + 0 = cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) = Y + 0 >= Y + 0 = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = N + Y + 0 >= N + Y + 0 = cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) = 4N + 4 >= 4N + 4 = cons(N,n__nats(n__s(N))) zprimes() = 9 >= 4 = sieve(nats(s(s(0())))) filter(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = n__filter(X1,X2,X3) sieve(X) = X + 0 >= X = n__sieve(X) nats(X) = 4X + 4 >= 4X + 4 = n__nats(X) s(X) = X + 0 >= X + 0 = n__s(X) activate(n__filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) = X + 0 >= X + 0 = sieve(activate(X)) activate(n__nats(X)) = 4X + 4 >= 4X + 4 = nats(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(X) = X + 0 >= X = X problem: DPs: activate#(n__filter(X1,X2,X3)) -> activate#(X3) activate#(n__filter(X1,X2,X3)) -> activate#(X2) activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) -> activate#(Y) activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = x0 + 0, [filter#](x0, x1, x2) = x0 + x1 + 1x2 + 0, [zprimes] = 9, [n__nats](x0) = x0 + 2, [n__s](x0) = x0, [nats](x0) = x0 + 2, [n__sieve](x0) = 0, [sieve](x0) = 0, [s](x0) = x0 + 0, [n__filter](x0, x1, x2) = x0 + 1x1 + 1x2 + 3, [activate](x0) = x0 + 2, [filter](x0, x1, x2) = x0 + 1x1 + 1x2 + 3, [0] = 4, [cons](x0, x1) = x1 + 0 orientation: activate#(n__filter(X1,X2,X3)) = X1 + 1X2 + 1X3 + 3 >= X3 + 0 = activate#(X3) activate#(n__filter(X1,X2,X3)) = X1 + 1X2 + 1X3 + 3 >= X2 + 0 = activate#(X2) activate#(n__filter(X1,X2,X3)) = X1 + 1X2 + 1X3 + 3 >= X1 + 0 = activate#(X1) activate#(n__filter(X1,X2,X3)) = X1 + 1X2 + 1X3 + 3 >= X1 + X2 + 1X3 + 3 = filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) = 1M + Y + 4 >= Y + 0 = activate#(Y) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) filter#(cons(X,Y),s(N),M) = 1M + N + Y + 0 >= Y + 0 = activate#(Y) filter(cons(X,Y),0(),M) = 1M + Y + 5 >= 1M + Y + 3 = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = 1M + 1N + Y + 3 >= 1M + 1N + Y + 3 = cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) = 0 >= 0 = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = 0 >= 0 = cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) = N + 2 >= N + 2 = cons(N,n__nats(n__s(N))) zprimes() = 9 >= 0 = sieve(nats(s(s(0())))) filter(X1,X2,X3) = X1 + 1X2 + 1X3 + 3 >= X1 + 1X2 + 1X3 + 3 = n__filter(X1,X2,X3) sieve(X) = 0 >= 0 = n__sieve(X) nats(X) = X + 2 >= X + 2 = n__nats(X) s(X) = X + 0 >= X = n__s(X) activate(n__filter(X1,X2,X3)) = X1 + 1X2 + 1X3 + 3 >= X1 + 1X2 + 1X3 + 3 = filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) = 2 >= 0 = sieve(activate(X)) activate(n__nats(X)) = X + 2 >= X + 2 = nats(activate(X)) activate(n__s(X)) = X + 2 >= X + 2 = s(activate(X)) activate(X) = X + 2 >= X = X problem: DPs: activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) -> activate#(Y) activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = 4x0, [filter#](x0, x1, x2) = 4x0, [zprimes] = 9, [n__nats](x0) = 0, [n__s](x0) = 1x0, [nats](x0) = 0, [n__sieve](x0) = 0, [sieve](x0) = 0, [s](x0) = 1x0, [n__filter](x0, x1, x2) = x0, [activate](x0) = x0, [filter](x0, x1, x2) = x0, [0] = 0, [cons](x0, x1) = x1 orientation: activate#(n__filter(X1,X2,X3)) = 4X1 >= 4X1 = activate#(X1) activate#(n__filter(X1,X2,X3)) = 4X1 >= 4X1 = filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) = 4Y >= 4Y = activate#(Y) activate#(n__s(X)) = 5X >= 4X = activate#(X) filter#(cons(X,Y),s(N),M) = 4Y >= 4Y = activate#(Y) filter(cons(X,Y),0(),M) = Y >= Y = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = Y >= Y = cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) = 0 >= 0 = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = 0 >= 0 = cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) = 0 >= 0 = cons(N,n__nats(n__s(N))) zprimes() = 9 >= 0 = sieve(nats(s(s(0())))) filter(X1,X2,X3) = X1 >= X1 = n__filter(X1,X2,X3) sieve(X) = 0 >= 0 = n__sieve(X) nats(X) = 0 >= 0 = n__nats(X) s(X) = 1X >= 1X = n__s(X) activate(n__filter(X1,X2,X3)) = X1 >= X1 = filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) = 0 >= 0 = sieve(activate(X)) activate(n__nats(X)) = 0 >= 0 = nats(activate(X)) activate(n__s(X)) = 1X >= 1X = s(activate(X)) activate(X) = X >= X = X problem: DPs: activate#(n__filter(X1,X2,X3)) -> activate#(X1) activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) -> activate#(Y) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = x0, [filter#](x0, x1, x2) = x0 + x1 + x2 + 0, [zprimes] = 0, [n__nats](x0) = x0 + 0, [n__s](x0) = x0, [nats](x0) = x0 + 0, [n__sieve](x0) = 0, [sieve](x0) = 0, [s](x0) = x0, [n__filter](x0, x1, x2) = 1x0 + x1 + x2 + 0, [activate](x0) = x0, [filter](x0, x1, x2) = 1x0 + x1 + x2 + 0, [0] = 0, [cons](x0, x1) = x1 orientation: activate#(n__filter(X1,X2,X3)) = 1X1 + X2 + X3 + 0 >= X1 = activate#(X1) activate#(n__filter(X1,X2,X3)) = 1X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) = M + Y + 0 >= Y = activate#(Y) filter#(cons(X,Y),s(N),M) = M + N + Y + 0 >= Y = activate#(Y) filter(cons(X,Y),0(),M) = M + 1Y + 0 >= M + 1Y + 0 = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = M + N + 1Y + 0 >= M + N + 1Y + 0 = cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) = 0 >= 0 = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = 0 >= 0 = cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) = N + 0 >= N + 0 = cons(N,n__nats(n__s(N))) zprimes() = 0 >= 0 = sieve(nats(s(s(0())))) filter(X1,X2,X3) = 1X1 + X2 + X3 + 0 >= 1X1 + X2 + X3 + 0 = n__filter(X1,X2,X3) sieve(X) = 0 >= 0 = n__sieve(X) nats(X) = X + 0 >= X + 0 = n__nats(X) s(X) = X >= X = n__s(X) activate(n__filter(X1,X2,X3)) = 1X1 + X2 + X3 + 0 >= 1X1 + X2 + X3 + 0 = filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) = 0 >= 0 = sieve(activate(X)) activate(n__nats(X)) = X + 0 >= X + 0 = nats(activate(X)) activate(n__s(X)) = X >= X = s(activate(X)) activate(X) = X >= X = X problem: DPs: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) -> activate#(Y) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = x0 + 0, [filter#](x0, x1, x2) = 1x0 + 0, [zprimes] = 5, [n__nats](x0) = 2, [n__s](x0) = 4, [nats](x0) = 2, [n__sieve](x0) = 3, [sieve](x0) = 3, [s](x0) = 4, [n__filter](x0, x1, x2) = 1x0, [activate](x0) = x0, [filter](x0, x1, x2) = 1x0, [0] = 5, [cons](x0, x1) = x1 + 1 orientation: activate#(n__filter(X1,X2,X3)) = 1X1 + 0 >= 1X1 + 0 = filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) = 1Y + 2 >= Y + 0 = activate#(Y) filter#(cons(X,Y),s(N),M) = 1Y + 2 >= Y + 0 = activate#(Y) filter(cons(X,Y),0(),M) = 1Y + 2 >= 1Y + 1 = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = 1Y + 2 >= 1Y + 1 = cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) = 3 >= 3 = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = 3 >= 3 = cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) = 2 >= 2 = cons(N,n__nats(n__s(N))) zprimes() = 5 >= 3 = sieve(nats(s(s(0())))) filter(X1,X2,X3) = 1X1 >= 1X1 = n__filter(X1,X2,X3) sieve(X) = 3 >= 3 = n__sieve(X) nats(X) = 2 >= 2 = n__nats(X) s(X) = 4 >= 4 = n__s(X) activate(n__filter(X1,X2,X3)) = 1X1 >= 1X1 = filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) = 3 >= 3 = sieve(activate(X)) activate(n__nats(X)) = 2 >= 2 = nats(activate(X)) activate(n__s(X)) = 4 >= 4 = s(activate(X)) activate(X) = X >= X = X problem: DPs: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3)) TRS: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) nats(N) -> cons(N,n__nats(n__s(N))) zprimes() -> sieve(nats(s(s(0())))) filter(X1,X2,X3) -> n__filter(X1,X2,X3) sieve(X) -> n__sieve(X) nats(X) -> n__nats(X) s(X) -> n__s(X) activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) -> sieve(activate(X)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 44/1