/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- 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 Usable Rule Processor: 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: 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 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)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) 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))) sieve(X) -> n__sieve(X) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) Arctic Interpretation Processor: dimension: 1 usable rules: 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 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)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) 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))) sieve(X) -> n__sieve(X) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) interpretation: [sieve#](x0) = x0 + 0, [activate#](x0) = x0 + 0, [filter#](x0, x1, x2) = x0 + x1 + x2 + 0, [n__nats](x0) = x0, [n__s](x0) = x0, [nats](x0) = x0 + 0, [n__sieve](x0) = 2x0 + 2, [sieve](x0) = 2x0 + 2, [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 orientation: sieve#(cons(0(),Y)) = Y + 0 >= Y + 0 = activate#(Y) 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__sieve(X)) = 2X + 2 >= X + 0 = activate#(X) activate#(n__sieve(X)) = 2X + 2 >= X + 0 = sieve#(activate(X)) sieve#(cons(s(N),Y)) = N + Y + 0 >= Y + 0 = activate#(Y) activate#(n__nats(X)) = X + 0 >= X + 0 = activate#(X) 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) 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)) = 2X + 2 >= 2X + 2 = 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 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)) filter(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = n__filter(X1,X2,X3) sieve(cons(0(),Y)) = 2Y + 2 >= 2Y + 2 = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = 2N + 2Y + 2 >= 2N + 2Y + 2 = cons(s(N),n__sieve(n__filter(activate(Y),N,N))) sieve(X) = 2X + 2 >= 2X + 2 = n__sieve(X) nats(N) = N + 0 >= N = cons(N,n__nats(n__s(N))) nats(X) = X + 0 >= X = n__nats(X) s(X) = X + 0 >= X = n__s(X) problem: 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) 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: 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 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)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) 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))) sieve(X) -> n__sieve(X) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) Restore Modifier: 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) 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 SCC Processor: #sccs: 1 #rules: 8 #arcs: 84/100 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__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 Usable Rule Processor: 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__nats(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: 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 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)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) 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))) sieve(X) -> n__sieve(X) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) Arctic Interpretation Processor: dimension: 1 usable rules: 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 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)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) 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))) sieve(X) -> n__sieve(X) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) interpretation: [activate#](x0) = x0 + 4, [filter#](x0, x1, x2) = 3x0 + 0, [n__nats](x0) = x0 + 1, [n__s](x0) = x0, [nats](x0) = x0 + 1, [n__sieve](x0) = 1, [sieve](x0) = 1, [s](x0) = x0 + 0, [n__filter](x0, x1, x2) = 4x0 + x1 + 4x2 + 4, [activate](x0) = x0 + 0, [filter](x0, x1, x2) = 4x0 + x1 + 4x2 + 4, [0] = 0, [cons](x0, x1) = x1 + 1 orientation: activate#(n__filter(X1,X2,X3)) = 4X1 + X2 + 4X3 + 4 >= X3 + 4 = activate#(X3) activate#(n__filter(X1,X2,X3)) = 4X1 + X2 + 4X3 + 4 >= X2 + 4 = activate#(X2) activate#(n__filter(X1,X2,X3)) = 4X1 + X2 + 4X3 + 4 >= X1 + 4 = activate#(X1) activate#(n__filter(X1,X2,X3)) = 4X1 + X2 + 4X3 + 4 >= 3X1 + 3 = filter#(activate(X1),activate(X2),activate(X3)) filter#(cons(X,Y),0(),M) = 3Y + 4 >= Y + 4 = activate#(Y) activate#(n__nats(X)) = X + 4 >= X + 4 = activate#(X) activate#(n__s(X)) = X + 4 >= X + 4 = activate#(X) filter#(cons(X,Y),s(N),M) = 3Y + 4 >= Y + 4 = activate#(Y) activate(n__filter(X1,X2,X3)) = 4X1 + X2 + 4X3 + 4 >= 4X1 + X2 + 4X3 + 4 = filter(activate(X1),activate(X2),activate(X3)) activate(n__sieve(X)) = 1 >= 1 = sieve(activate(X)) activate(n__nats(X)) = X + 1 >= X + 1 = nats(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(X) = X + 0 >= X = X filter(cons(X,Y),0(),M) = 4M + 4Y + 5 >= 4M + 4Y + 4 = cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) = 4M + N + 4Y + 5 >= 4M + N + 4Y + 4 = cons(X,n__filter(activate(Y),N,M)) filter(X1,X2,X3) = 4X1 + X2 + 4X3 + 4 >= 4X1 + X2 + 4X3 + 4 = n__filter(X1,X2,X3) sieve(cons(0(),Y)) = 1 >= 1 = cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) = 1 >= 1 = cons(s(N),n__sieve(n__filter(activate(Y),N,N))) sieve(X) = 1 >= 1 = n__sieve(X) nats(N) = N + 1 >= N + 1 = cons(N,n__nats(n__s(N))) nats(X) = X + 1 >= X + 1 = n__nats(X) s(X) = X + 0 >= X = n__s(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) filter#(cons(X,Y),0(),M) -> activate#(Y) activate#(n__nats(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) filter#(cons(X,Y),s(N),M) -> activate#(Y) TRS: 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 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)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) 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))) sieve(X) -> n__sieve(X) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) Restore Modifier: 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) filter#(cons(X,Y),0(),M) -> 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 SCC Processor: #sccs: 1 #rules: 5 #arcs: 44/49 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__s(X)) -> 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 Size-Change Termination Processor: DPs: 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 The DP: activate#(n__nats(X)) -> activate#(X) has the edges: 0 > 0 The DP: activate#(n__filter(X1,X2,X3)) -> activate#(X3) has the edges: 0 > 0 The DP: activate#(n__filter(X1,X2,X3)) -> activate#(X2) has the edges: 0 > 0 The DP: activate#(n__filter(X1,X2,X3)) -> activate#(X1) has the edges: 0 > 0 The DP: activate#(n__s(X)) -> activate#(X) has the edges: 0 > 0 Qed