MAYBE Time: 0.862072 TRS: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} DP: DP: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2), filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3), active# cons(X1, X2) -> cons#(active X1, X2), active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> filter#(X1, X2, active X3), active# filter(X1, X2, X3) -> filter#(X1, active X2, X3), active# filter(X1, X2, X3) -> filter#(active X1, X2, X3), active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3, active# filter(cons(X, Y), 0(), M) -> cons#(0(), filter(Y, M, M)), active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M), active# filter(cons(X, Y), s N, M) -> cons#(X, filter(Y, N, M)), active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M), active# s X -> active# X, active# s X -> s# active X, active# sieve X -> active# X, active# sieve X -> sieve# active X, active# sieve cons(0(), Y) -> cons#(0(), sieve Y), active# sieve cons(0(), Y) -> sieve# Y, active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N)), active# sieve cons(s N, Y) -> filter#(Y, N, N), active# sieve cons(s N, Y) -> sieve# filter(Y, N, N), active# nats X -> active# X, active# nats X -> nats# active X, active# nats N -> cons#(N, nats s N), active# nats N -> s# N, active# nats N -> nats# s N, active# zprimes() -> s# 0(), active# zprimes() -> s# s 0(), active# zprimes() -> sieve# nats s s 0(), active# zprimes() -> nats# s s 0(), s# mark X -> s# X, s# ok X -> s# X, sieve# mark X -> sieve# X, sieve# ok X -> sieve# X, nats# mark X -> nats# X, nats# ok X -> nats# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2), proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> filter#(proper X1, proper X2, proper X3), proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3, proper# s X -> s# proper X, proper# s X -> proper# X, proper# sieve X -> sieve# proper X, proper# sieve X -> proper# X, proper# nats X -> nats# proper X, proper# nats X -> proper# X, top# mark X -> proper# X, top# mark X -> top# proper X, top# ok X -> active# X, top# ok X -> top# active X} TRS: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} EDG: { (cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (cons#(mark X1, X2) -> cons#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)) (active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3)) (active# sieve cons(s N, Y) -> sieve# filter(Y, N, N), sieve# ok X -> sieve# X) (active# sieve cons(s N, Y) -> sieve# filter(Y, N, N), sieve# mark X -> sieve# X) (active# sieve cons(0(), Y) -> sieve# Y, sieve# ok X -> sieve# X) (active# sieve cons(0(), Y) -> sieve# Y, sieve# mark X -> sieve# X) (active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N)), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N)), cons#(mark X1, X2) -> cons#(X1, X2)) (filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)) (filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3)) (filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3)) (filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(X1, X2, mark X3) -> filter#(X1, X2, X3)) (filter#(mark X1, X2, X3) -> filter#(X1, X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)) (filter#(mark X1, X2, X3) -> filter#(X1, X2, X3), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3)) (filter#(mark X1, X2, X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3)) (filter#(mark X1, X2, X3) -> filter#(X1, X2, X3), filter#(X1, X2, mark X3) -> filter#(X1, X2, X3)) (active# filter(X1, X2, X3) -> filter#(active X1, X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)) (active# filter(X1, X2, X3) -> filter#(active X1, X2, X3), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3)) (active# sieve X -> active# X, active# zprimes() -> nats# s s 0()) (active# sieve X -> active# X, active# zprimes() -> sieve# nats s s 0()) (active# sieve X -> active# X, active# zprimes() -> s# s 0()) (active# sieve X -> active# X, active# zprimes() -> s# 0()) (active# sieve X -> active# X, active# nats N -> nats# s N) (active# sieve X -> active# X, active# nats N -> s# N) (active# sieve X -> active# X, active# nats N -> cons#(N, nats s N)) (active# sieve X -> active# X, active# nats X -> nats# active X) (active# sieve X -> active# X, active# nats X -> active# X) (active# sieve X -> active# X, active# sieve cons(s N, Y) -> sieve# filter(Y, N, N)) (active# sieve X -> active# X, active# sieve cons(s N, Y) -> filter#(Y, N, N)) (active# sieve X -> active# X, active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N))) (active# sieve X -> active# X, active# sieve cons(0(), Y) -> sieve# Y) (active# sieve X -> active# X, active# sieve cons(0(), Y) -> cons#(0(), sieve Y)) (active# sieve X -> active# X, active# sieve X -> sieve# active X) (active# sieve X -> active# X, active# sieve X -> active# X) (active# sieve X -> active# X, active# s X -> s# active X) (active# sieve X -> active# X, active# s X -> active# X) (active# sieve X -> active# X, active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M)) (active# sieve X -> active# X, active# filter(cons(X, Y), s N, M) -> cons#(X, filter(Y, N, M))) (active# sieve X -> active# X, active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M)) (active# sieve X -> active# X, active# filter(cons(X, Y), 0(), M) -> cons#(0(), filter(Y, M, M))) (active# sieve X -> active# X, active# filter(X1, X2, X3) -> active# X3) (active# sieve X -> active# X, active# filter(X1, X2, X3) -> active# X2) (active# sieve X -> active# X, active# filter(X1, X2, X3) -> active# X1) (active# sieve X -> active# X, active# filter(X1, X2, X3) -> filter#(active X1, X2, X3)) (active# sieve X -> active# X, active# filter(X1, X2, X3) -> filter#(X1, active X2, X3)) (active# sieve X -> active# X, active# filter(X1, X2, X3) -> filter#(X1, X2, active X3)) (active# sieve X -> active# X, active# cons(X1, X2) -> active# X1) (active# sieve X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (s# mark X -> s# X, s# ok X -> s# X) (s# mark X -> s# X, s# mark X -> s# X) (sieve# mark X -> sieve# X, sieve# ok X -> sieve# X) (sieve# mark X -> sieve# X, sieve# mark X -> sieve# X) (nats# mark X -> nats# X, nats# ok X -> nats# X) (nats# mark X -> nats# X, nats# mark X -> nats# X) (proper# s X -> proper# X, proper# nats X -> proper# X) (proper# s X -> proper# X, proper# nats X -> nats# proper X) (proper# s X -> proper# X, proper# sieve X -> proper# X) (proper# s X -> proper# X, proper# sieve X -> sieve# proper X) (proper# s X -> proper# X, proper# s X -> proper# X) (proper# s X -> proper# X, proper# s X -> s# proper X) (proper# s X -> proper# X, proper# filter(X1, X2, X3) -> proper# X3) (proper# s X -> proper# X, proper# filter(X1, X2, X3) -> proper# X2) (proper# s X -> proper# X, proper# filter(X1, X2, X3) -> proper# X1) (proper# s X -> proper# X, proper# filter(X1, X2, X3) -> filter#(proper X1, proper X2, proper X3)) (proper# s X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# nats X -> proper# X, proper# nats X -> proper# X) (proper# nats X -> proper# X, proper# nats X -> nats# proper X) (proper# nats X -> proper# X, proper# sieve X -> proper# X) (proper# nats X -> proper# X, proper# sieve X -> sieve# proper X) (proper# nats X -> proper# X, proper# s X -> proper# X) (proper# nats X -> proper# X, proper# s X -> s# proper X) (proper# nats X -> proper# X, proper# filter(X1, X2, X3) -> proper# X3) (proper# nats X -> proper# X, proper# filter(X1, X2, X3) -> proper# X2) (proper# nats X -> proper# X, proper# filter(X1, X2, X3) -> proper# X1) (proper# nats X -> proper# X, proper# filter(X1, X2, X3) -> filter#(proper X1, proper X2, proper X3)) (proper# nats X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# nats X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# nats X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (top# ok X -> active# X, active# zprimes() -> nats# s s 0()) (top# ok X -> active# X, active# zprimes() -> sieve# nats s s 0()) (top# ok X -> active# X, active# zprimes() -> s# s 0()) (top# ok X -> active# X, active# zprimes() -> s# 0()) (top# ok X -> active# X, active# nats N -> nats# s N) (top# ok X -> active# X, active# nats N -> s# N) (top# ok X -> active# X, active# nats N -> cons#(N, nats s N)) (top# ok X -> active# X, active# nats X -> nats# active X) (top# ok X -> active# X, active# nats X -> active# X) (top# ok X -> active# X, active# sieve cons(s N, Y) -> sieve# filter(Y, N, N)) (top# ok X -> active# X, active# sieve cons(s N, Y) -> filter#(Y, N, N)) (top# ok X -> active# X, active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N))) (top# ok X -> active# X, active# sieve cons(0(), Y) -> sieve# Y) (top# ok X -> active# X, active# sieve cons(0(), Y) -> cons#(0(), sieve Y)) (top# ok X -> active# X, active# sieve X -> sieve# active X) (top# ok X -> active# X, active# sieve X -> active# X) (top# ok X -> active# X, active# s X -> s# active X) (top# ok X -> active# X, active# s X -> active# X) (top# ok X -> active# X, active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M)) (top# ok X -> active# X, active# filter(cons(X, Y), s N, M) -> cons#(X, filter(Y, N, M))) (top# ok X -> active# X, active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M)) (top# ok X -> active# X, active# filter(cons(X, Y), 0(), M) -> cons#(0(), filter(Y, M, M))) (top# ok X -> active# X, active# filter(X1, X2, X3) -> active# X3) (top# ok X -> active# X, active# filter(X1, X2, X3) -> active# X2) (top# ok X -> active# X, active# filter(X1, X2, X3) -> active# X1) (top# ok X -> active# X, active# filter(X1, X2, X3) -> filter#(active X1, X2, X3)) (top# ok X -> active# X, active# filter(X1, X2, X3) -> filter#(X1, active X2, X3)) (top# ok X -> active# X, active# filter(X1, X2, X3) -> filter#(X1, X2, active X3)) (top# ok X -> active# X, active# cons(X1, X2) -> active# X1) (top# ok X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# cons(X1, X2) -> active# X1, active# zprimes() -> nats# s s 0()) (active# cons(X1, X2) -> active# X1, active# zprimes() -> sieve# nats s s 0()) (active# cons(X1, X2) -> active# X1, active# zprimes() -> s# s 0()) (active# cons(X1, X2) -> active# X1, active# zprimes() -> s# 0()) (active# cons(X1, X2) -> active# X1, active# nats N -> nats# s N) (active# cons(X1, X2) -> active# X1, active# nats N -> s# N) (active# cons(X1, X2) -> active# X1, active# nats N -> cons#(N, nats s N)) (active# cons(X1, X2) -> active# X1, active# nats X -> nats# active X) (active# cons(X1, X2) -> active# X1, active# nats X -> active# X) (active# cons(X1, X2) -> active# X1, active# sieve cons(s N, Y) -> sieve# filter(Y, N, N)) (active# cons(X1, X2) -> active# X1, active# sieve cons(s N, Y) -> filter#(Y, N, N)) (active# cons(X1, X2) -> active# X1, active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N))) (active# cons(X1, X2) -> active# X1, active# sieve cons(0(), Y) -> sieve# Y) (active# cons(X1, X2) -> active# X1, active# sieve cons(0(), Y) -> cons#(0(), sieve Y)) (active# cons(X1, X2) -> active# X1, active# sieve X -> sieve# active X) (active# cons(X1, X2) -> active# X1, active# sieve X -> active# X) (active# cons(X1, X2) -> active# X1, active# s X -> s# active X) (active# cons(X1, X2) -> active# X1, active# s X -> active# X) (active# cons(X1, X2) -> active# X1, active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M)) (active# cons(X1, X2) -> active# X1, active# filter(cons(X, Y), s N, M) -> cons#(X, filter(Y, N, M))) (active# cons(X1, X2) -> active# X1, active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M)) (active# cons(X1, X2) -> active# X1, active# filter(cons(X, Y), 0(), M) -> cons#(0(), filter(Y, M, M))) (active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X3) (active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X2) (active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1) (active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> filter#(active X1, X2, X3)) (active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> filter#(X1, active X2, X3)) (active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> filter#(X1, X2, active X3)) (active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> active# X1) (active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (proper# cons(X1, X2) -> proper# X1, proper# nats X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# nats X -> nats# proper X) (proper# cons(X1, X2) -> proper# X1, proper# sieve X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# sieve X -> sieve# proper X) (proper# cons(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# cons(X1, X2) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X3) (proper# cons(X1, X2) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2) (proper# cons(X1, X2) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X1) (proper# cons(X1, X2) -> proper# X1, proper# filter(X1, X2, X3) -> filter#(proper X1, proper X2, proper X3)) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (active# filter(X1, X2, X3) -> active# X3, active# zprimes() -> nats# s s 0()) (active# filter(X1, X2, X3) -> active# X3, active# zprimes() -> sieve# nats s s 0()) (active# filter(X1, X2, X3) -> active# X3, active# zprimes() -> s# s 0()) (active# filter(X1, X2, X3) -> active# X3, active# zprimes() -> s# 0()) (active# filter(X1, X2, X3) -> active# X3, active# nats N -> nats# s N) (active# filter(X1, X2, X3) -> active# X3, active# nats N -> s# N) (active# filter(X1, X2, X3) -> active# X3, active# nats N -> cons#(N, nats s N)) (active# filter(X1, X2, X3) -> active# X3, active# nats X -> nats# active X) (active# filter(X1, X2, X3) -> active# X3, active# nats X -> active# X) (active# filter(X1, X2, X3) -> active# X3, active# sieve cons(s N, Y) -> sieve# filter(Y, N, N)) (active# filter(X1, X2, X3) -> active# X3, active# sieve cons(s N, Y) -> filter#(Y, N, N)) (active# filter(X1, X2, X3) -> active# X3, active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N))) (active# filter(X1, X2, X3) -> active# X3, active# sieve cons(0(), Y) -> sieve# Y) (active# filter(X1, X2, X3) -> active# X3, active# sieve cons(0(), Y) -> cons#(0(), sieve Y)) (active# filter(X1, X2, X3) -> active# X3, active# sieve X -> sieve# active X) (active# filter(X1, X2, X3) -> active# X3, active# sieve X -> active# X) (active# filter(X1, X2, X3) -> active# X3, active# s X -> s# active X) (active# filter(X1, X2, X3) -> active# X3, active# s X -> active# X) (active# filter(X1, X2, X3) -> active# X3, active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M)) (active# filter(X1, X2, X3) -> active# X3, active# filter(cons(X, Y), s N, M) -> cons#(X, filter(Y, N, M))) (active# filter(X1, X2, X3) -> active# X3, active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M)) (active# filter(X1, X2, X3) -> active# X3, active# filter(cons(X, Y), 0(), M) -> cons#(0(), filter(Y, M, M))) (active# filter(X1, X2, X3) -> active# X3, active# filter(X1, X2, X3) -> active# X3) (active# filter(X1, X2, X3) -> active# X3, active# filter(X1, X2, X3) -> active# X2) (active# filter(X1, X2, X3) -> active# X3, active# filter(X1, X2, X3) -> active# X1) (active# filter(X1, X2, X3) -> active# X3, active# filter(X1, X2, X3) -> filter#(active X1, X2, X3)) (active# filter(X1, X2, X3) -> active# X3, active# filter(X1, X2, X3) -> filter#(X1, active X2, X3)) (active# filter(X1, X2, X3) -> active# X3, active# filter(X1, X2, X3) -> filter#(X1, X2, active X3)) (active# filter(X1, X2, X3) -> active# X3, active# cons(X1, X2) -> active# X1) (active# filter(X1, X2, X3) -> active# X3, active# cons(X1, X2) -> cons#(active X1, X2)) (active# nats N -> nats# s N, nats# ok X -> nats# X) (active# nats N -> nats# s N, nats# mark X -> nats# X) (active# s X -> s# active X, s# ok X -> s# X) (active# s X -> s# active X, s# mark X -> s# X) (active# nats X -> nats# active X, nats# ok X -> nats# X) (active# nats X -> nats# active X, nats# mark X -> nats# X) (proper# sieve X -> sieve# proper X, sieve# ok X -> sieve# X) (proper# sieve X -> sieve# proper X, sieve# mark X -> sieve# X) (top# mark X -> top# proper X, top# ok X -> top# active X) (top# mark X -> top# proper X, top# ok X -> active# X) (top# mark X -> top# 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filter#(X1, X2, active X3), filter#(X1, X2, mark X3) -> filter#(X1, X2, X3)) (active# filter(X1, X2, X3) -> filter#(X1, X2, active X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)) (active# sieve cons(s N, Y) -> filter#(Y, N, N), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3)) (proper# filter(X1, X2, X3) -> proper# X3, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# filter(X1, X2, X3) -> proper# X3, proper# cons(X1, X2) -> proper# X1) (proper# filter(X1, X2, X3) -> proper# X3, proper# cons(X1, X2) -> proper# X2) (proper# filter(X1, X2, X3) -> proper# X3, proper# filter(X1, X2, X3) -> filter#(proper X1, proper X2, proper X3)) (proper# filter(X1, X2, X3) -> proper# X3, proper# filter(X1, X2, X3) -> proper# X1) (proper# filter(X1, X2, X3) -> proper# X3, proper# filter(X1, X2, X3) -> proper# X2) (proper# filter(X1, X2, X3) -> proper# X3, proper# filter(X1, X2, X3) -> proper# X3) (proper# filter(X1, X2, X3) -> proper# X3, proper# s X -> s# proper X) (proper# 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filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2) (active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X3) (active# filter(X1, X2, X3) -> active# X1, active# filter(cons(X, Y), 0(), M) -> cons#(0(), filter(Y, M, M))) (active# filter(X1, X2, X3) -> active# X1, active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M)) (active# filter(X1, X2, X3) -> active# X1, active# filter(cons(X, Y), s N, M) -> cons#(X, filter(Y, N, M))) (active# filter(X1, X2, X3) -> active# X1, active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M)) (active# filter(X1, X2, X3) -> active# X1, active# s X -> active# X) (active# filter(X1, X2, X3) -> active# X1, active# s X -> s# active X) (active# filter(X1, X2, X3) -> active# X1, active# sieve X -> active# X) (active# filter(X1, X2, X3) -> active# X1, active# sieve X -> sieve# active X) (active# filter(X1, X2, X3) -> active# X1, active# sieve cons(0(), Y) -> cons#(0(), sieve Y)) (active# filter(X1, X2, X3) -> 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active# X, active# filter(cons(X, Y), s N, M) -> cons#(X, filter(Y, N, M))) (active# nats X -> active# X, active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M)) (active# nats X -> active# X, active# s X -> active# X) (active# nats X -> active# X, active# s X -> s# active X) (active# nats X -> active# X, active# sieve X -> active# X) (active# nats X -> active# X, active# sieve X -> sieve# active X) (active# nats X -> active# X, active# sieve cons(0(), Y) -> cons#(0(), sieve Y)) (active# nats X -> active# X, active# sieve cons(0(), Y) -> sieve# Y) (active# nats X -> active# X, active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N))) (active# nats X -> active# X, active# sieve cons(s N, Y) -> filter#(Y, N, N)) (active# nats X -> active# X, active# sieve cons(s N, Y) -> sieve# filter(Y, N, N)) (active# nats X -> active# X, active# nats X -> active# X) (active# nats X -> active# X, active# nats X -> nats# active X) (active# nats X -> active# X, active# nats N -> cons#(N, nats s N)) (active# nats X -> active# X, active# nats N -> s# N) (active# nats X -> active# X, active# nats N -> nats# s N) (active# nats X -> active# X, active# zprimes() -> s# 0()) (active# nats X -> active# X, active# zprimes() -> s# s 0()) (active# nats X -> active# X, active# zprimes() -> sieve# nats s s 0()) (active# nats X -> active# X, active# zprimes() -> nats# s s 0()) (active# s X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# s X -> active# X, active# cons(X1, X2) -> active# X1) (active# s X -> active# X, active# filter(X1, X2, X3) -> filter#(X1, X2, active X3)) (active# s X -> active# X, active# filter(X1, X2, X3) -> filter#(X1, active X2, X3)) (active# s X -> active# X, active# filter(X1, X2, X3) -> filter#(active X1, X2, X3)) (active# s X -> active# X, active# filter(X1, X2, X3) -> active# X1) (active# s X -> active# X, active# filter(X1, X2, X3) -> active# X2) (active# s X -> active# X, active# filter(X1, X2, X3) -> active# X3) (active# s X -> active# X, active# filter(cons(X, Y), 0(), M) -> cons#(0(), filter(Y, M, M))) (active# s X -> active# X, active# filter(cons(X, Y), 0(), M) -> filter#(Y, M, M)) (active# s X -> active# X, active# filter(cons(X, Y), s N, M) -> cons#(X, filter(Y, N, M))) (active# s X -> active# X, active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M)) (active# s X -> active# X, active# s X -> active# X) (active# s X -> active# X, active# s X -> s# active X) (active# s X -> active# X, active# sieve X -> active# X) (active# s X -> active# X, active# sieve X -> sieve# active X) (active# s X -> active# X, active# sieve cons(0(), Y) -> cons#(0(), sieve Y)) (active# s X -> active# X, active# sieve cons(0(), Y) -> sieve# Y) (active# s X -> active# X, active# sieve cons(s N, Y) -> cons#(s N, sieve filter(Y, N, N))) (active# s X -> active# X, active# sieve cons(s N, Y) -> filter#(Y, N, N)) (active# s X -> active# X, active# sieve cons(s N, Y) -> sieve# filter(Y, N, N)) (active# s X -> active# X, active# nats X -> active# X) (active# s X -> active# X, active# nats X -> nats# active X) (active# s X -> active# X, active# nats N -> cons#(N, nats s N)) (active# s X -> active# X, active# nats N -> s# N) (active# s X -> active# X, active# nats N -> nats# s N) (active# s X -> active# X, active# zprimes() -> s# 0()) (active# s X -> active# X, active# zprimes() -> s# s 0()) (active# s X -> active# X, active# zprimes() -> sieve# nats s s 0()) (active# s X -> active# X, active# zprimes() -> nats# s s 0()) (filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3), filter#(X1, X2, mark X3) -> filter#(X1, X2, X3)) (filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3)) (filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3)) (filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)) (filter#(X1, mark X2, X3) -> filter#(X1, X2, X3), filter#(X1, X2, mark X3) -> filter#(X1, X2, X3)) (filter#(X1, mark X2, X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3)) (filter#(X1, mark X2, X3) -> filter#(X1, X2, X3), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3)) (filter#(X1, mark X2, X3) -> filter#(X1, X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)) (active# filter(X1, X2, X3) -> filter#(X1, active X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3)) (active# filter(X1, X2, X3) -> filter#(X1, active X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)) (proper# cons(X1, X2) -> cons#(proper X1, proper X2), cons#(mark X1, X2) -> cons#(X1, X2)) (proper# cons(X1, X2) -> cons#(proper X1, proper X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# filter(cons(X, Y), s N, M) -> filter#(Y, N, M), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3)) (cons#(ok X1, ok X2) -> cons#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (cons#(ok X1, ok X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) } STATUS: arrows: 0.874749 SCCS (8): Scc: {top# mark X -> top# proper X, top# ok X -> top# active X} Scc: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# sieve X -> proper# X, proper# nats X -> proper# X} Scc: { active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3, active# s X -> active# X, active# sieve X -> active# X, active# nats X -> active# X} Scc: {nats# mark X -> nats# X, nats# ok X -> nats# X} Scc: {s# mark X -> s# X, s# ok X -> s# X} Scc: {sieve# mark X -> sieve# X, sieve# ok X -> sieve# X} Scc: { filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)} Scc: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)} SCC (2): Strict: {top# mark X -> top# proper X, top# ok X -> top# active X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} Fail SCC (8): Strict: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# sieve X -> proper# X, proper# nats X -> proper# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + x1 + x2, [cons](x0, x1) = x0 + x1, [mark](x0) = 0, [active](x0) = 0, [s](x0) = x0, [sieve](x0) = x0, [nats](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [zprimes] = 1, [proper#](x0) = x0 Strict: proper# nats X -> proper# X 1 + 1X >= 0 + 1X proper# sieve X -> proper# X 0 + 1X >= 0 + 1X proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# filter(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# filter(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# filter(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 1 + 0X >= 0 + 0X sieve ok X -> ok sieve X 1 + 1X >= 1 + 1X sieve mark X -> mark sieve X 0 + 0X >= 0 + 0X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X active zprimes() -> mark sieve nats s s 0() 0 >= 0 active nats N -> mark cons(N, nats s N) 0 + 0N >= 0 + 0N active nats X -> nats active X 0 + 0X >= 1 + 0X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 0 + 0Y + 0N >= 0 + 0Y + 0N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 0 + 0Y >= 0 + 0Y active sieve X -> sieve active X 0 + 0X >= 0 + 0X active s X -> s active X 0 + 0X >= 0 + 0X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 0 + 0Y + 0M + 0X + 0N >= 0 + 0Y + 0M + 0X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 0 + 0Y + 0M + 0X >= 0 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# sieve X -> proper# X} SCC (7): Strict: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3, proper# s X -> proper# X, proper# sieve X -> proper# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + x1 + x2, [cons](x0, x1) = x0 + x1, [mark](x0) = 0, [active](x0) = 0, [s](x0) = x0, [sieve](x0) = x0 + 1, [nats](x0) = x0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [zprimes] = 1, [proper#](x0) = x0 Strict: proper# sieve X -> proper# X 1 + 1X >= 0 + 1X proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# filter(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# filter(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# filter(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 0 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 nats ok X -> ok nats X 1 + 1X >= 1 + 1X nats mark X -> mark nats X 0 + 0X >= 0 + 0X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 1 + 0X >= 0 + 0X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X active zprimes() -> mark sieve nats s s 0() 0 >= 0 active nats N -> mark cons(N, nats s N) 0 + 0N >= 0 + 0N active nats X -> nats active X 0 + 0X >= 0 + 0X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 0 + 0Y + 0N >= 0 + 0Y + 0N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 0 + 0Y >= 0 + 0Y active sieve X -> sieve active X 0 + 0X >= 1 + 0X active s X -> s active X 0 + 0X >= 0 + 0X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 0 + 0Y + 0M + 0X + 0N >= 0 + 0Y + 0M + 0X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 0 + 0Y + 0M + 0X >= 0 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3, proper# s X -> proper# X} SCC (6): Strict: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3, proper# s X -> proper# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + x1 + x2, [cons](x0, x1) = x0 + x1, [mark](x0) = 0, [active](x0) = 0, [s](x0) = x0 + 1, [sieve](x0) = 0, [nats](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [zprimes] = 1, [proper#](x0) = x0 Strict: proper# s X -> proper# X 1 + 1X >= 0 + 1X proper# filter(X1, X2, X3) -> proper# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 proper# filter(X1, X2, X3) -> proper# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 proper# filter(X1, X2, X3) -> proper# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 0 + 0X proper sieve X -> sieve proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 nats ok X -> ok nats X 0 + 0X >= 1 + 0X nats mark X -> mark nats X 0 + 0X >= 0 + 0X sieve ok X -> ok sieve X 0 + 0X >= 1 + 0X sieve mark X -> mark sieve X 0 + 0X >= 0 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 0X >= 0 + 0X active zprimes() -> mark sieve nats s s 0() 0 >= 0 active nats N -> mark cons(N, nats s N) 0 + 0N >= 0 + 0N active nats X -> nats active X 0 + 0X >= 0 + 0X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 0 + 0Y + 0N >= 0 + 0Y + 0N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 0 + 0Y >= 0 + 0Y active sieve X -> sieve active X 0 + 0X >= 0 + 0X active s X -> s active X 0 + 0X >= 1 + 0X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 0 + 0Y + 0M + 0X + 0N >= 0 + 0Y + 0M + 0X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 0 + 0Y + 0M + 0X >= 0 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3} SCC (5): Strict: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X1, proper# filter(X1, X2, X3) -> proper# X2, proper# filter(X1, X2, X3) -> proper# X3} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + x1 + x2 + 1, [cons](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = 1, [sieve](x0) = x0 + 1, [nats](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [zprimes] = 1, [proper#](x0) = x0 + 1 Strict: proper# filter(X1, X2, X3) -> proper# X3 2 + 1X1 + 1X2 + 1X3 >= 1 + 1X3 proper# filter(X1, X2, X3) -> proper# X2 2 + 1X1 + 1X2 + 1X3 >= 1 + 1X2 proper# filter(X1, X2, X3) -> proper# X1 2 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 proper# cons(X1, X2) -> proper# X2 2 + 1X1 + 1X2 >= 1 + 1X2 proper# cons(X1, X2) -> proper# X1 2 + 1X1 + 1X2 >= 1 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 nats ok X -> ok nats X 1 + 0X >= 2 + 0X nats mark X -> mark nats X 1 + 0X >= 2 + 0X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 2 + 0X active zprimes() -> mark sieve nats s s 0() 2 >= 3 active nats N -> mark cons(N, nats s N) 2 + 0N >= 3 + 1N active nats X -> nats active X 2 + 0X >= 1 + 0X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 4 + 1Y + 0N >= 5 + 1Y + 2N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 1Y >= 4 + 1Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 0X >= 1 + 0X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 4 + 1Y + 1M + 1X + 0N >= 3 + 1Y + 1M + 1X + 1N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 4 + 1Y + 1M + 1X >= 4 + 1Y + 2M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 4 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 Qed SCC (7): Strict: { active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3, active# s X -> active# X, active# sieve X -> active# X, active# nats X -> active# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + x1 + x2, [cons](x0, x1) = x0 + x1, [mark](x0) = 0, [active](x0) = 0, [s](x0) = x0, [sieve](x0) = x0, [nats](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [zprimes] = 1, [active#](x0) = x0 Strict: active# nats X -> active# X 1 + 1X >= 0 + 1X active# sieve X -> active# X 0 + 1X >= 0 + 1X active# s X -> active# X 0 + 1X >= 0 + 1X active# filter(X1, X2, X3) -> active# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 active# filter(X1, X2, X3) -> active# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 active# filter(X1, X2, X3) -> active# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 1 + 0X >= 0 + 0X sieve ok X -> ok sieve X 1 + 1X >= 1 + 1X sieve mark X -> mark sieve X 0 + 0X >= 0 + 0X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X active zprimes() -> mark sieve nats s s 0() 0 >= 0 active nats N -> mark cons(N, nats s N) 0 + 0N >= 0 + 0N active nats X -> nats active X 0 + 0X >= 1 + 0X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 0 + 0Y + 0N >= 0 + 0Y + 0N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 0 + 0Y >= 0 + 0Y active sieve X -> sieve active X 0 + 0X >= 0 + 0X active s X -> s active X 0 + 0X >= 0 + 0X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 0 + 0Y + 0M + 0X + 0N >= 0 + 0Y + 0M + 0X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 0 + 0Y + 0M + 0X >= 0 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: { active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3, active# s X -> active# X, active# sieve X -> active# X} SCC (6): Strict: { active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3, active# s X -> active# X, active# sieve X -> active# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + x1 + x2, [cons](x0, x1) = x0 + x1, [mark](x0) = 0, [active](x0) = 0, [s](x0) = x0, [sieve](x0) = x0 + 1, [nats](x0) = x0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [zprimes] = 1, [active#](x0) = x0 Strict: active# sieve X -> active# X 1 + 1X >= 0 + 1X active# s X -> active# X 0 + 1X >= 0 + 1X active# filter(X1, X2, X3) -> active# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 active# filter(X1, X2, X3) -> active# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 active# filter(X1, X2, X3) -> active# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 0 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 nats ok X -> ok nats X 1 + 1X >= 1 + 1X nats mark X -> mark nats X 0 + 0X >= 0 + 0X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 1 + 0X >= 0 + 0X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X active zprimes() -> mark sieve nats s s 0() 0 >= 0 active nats N -> mark cons(N, nats s N) 0 + 0N >= 0 + 0N active nats X -> nats active X 0 + 0X >= 0 + 0X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 0 + 0Y + 0N >= 0 + 0Y + 0N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 0 + 0Y >= 0 + 0Y active sieve X -> sieve active X 0 + 0X >= 1 + 0X active s X -> s active X 0 + 0X >= 0 + 0X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 0 + 0Y + 0M + 0X + 0N >= 0 + 0Y + 0M + 0X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 0 + 0Y + 0M + 0X >= 0 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: { active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3, active# s X -> active# X} SCC (5): Strict: { active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3, active# s X -> active# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + x1 + x2, [cons](x0, x1) = x0 + x1, [mark](x0) = 0, [active](x0) = 0, [s](x0) = x0 + 1, [sieve](x0) = 0, [nats](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [zprimes] = 1, [active#](x0) = x0 Strict: active# s X -> active# X 1 + 1X >= 0 + 1X active# filter(X1, X2, X3) -> active# X3 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X3 active# filter(X1, X2, X3) -> active# X2 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X2 active# filter(X1, X2, X3) -> active# X1 0 + 1X1 + 1X2 + 1X3 >= 0 + 1X1 active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 0 + 0X proper sieve X -> sieve proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 nats ok X -> ok nats X 0 + 0X >= 1 + 0X nats mark X -> mark nats X 0 + 0X >= 0 + 0X sieve ok X -> ok sieve X 0 + 0X >= 1 + 0X sieve mark X -> mark sieve X 0 + 0X >= 0 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 0X >= 0 + 0X active zprimes() -> mark sieve nats s s 0() 0 >= 0 active nats N -> mark cons(N, nats s N) 0 + 0N >= 0 + 0N active nats X -> nats active X 0 + 0X >= 0 + 0X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 0 + 0Y + 0N >= 0 + 0Y + 0N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 0 + 0Y >= 0 + 0Y active sieve X -> sieve active X 0 + 0X >= 0 + 0X active s X -> s active X 0 + 0X >= 1 + 0X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 0 + 0Y + 0M + 0X + 0N >= 0 + 0Y + 0M + 0X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 0 + 0Y + 0M + 0X >= 0 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 0X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 0 + 0X1 + 0X2 + 0X3 >= 0 + 1X1 + 1X2 + 0X3 active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 3 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 + 1X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 0 + 0X1 + 1X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 0X2 + 1X3 >= 0 + 0X1 + 0X2 + 0X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 0 + 1X1 + 1X2 + 0X3 >= 0 + 0X1 + 0X2 + 0X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: { active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3} SCC (4): Strict: { active# cons(X1, X2) -> active# X1, active# filter(X1, X2, X3) -> active# X1, active# filter(X1, X2, X3) -> active# X2, active# filter(X1, X2, X3) -> active# X3} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + x1 + x2 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = 1, [sieve](x0) = x0 + 1, [nats](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [zprimes] = 1, [active#](x0) = x0 + 1 Strict: active# filter(X1, X2, X3) -> active# X3 2 + 1X1 + 1X2 + 1X3 >= 1 + 1X3 active# filter(X1, X2, X3) -> active# X2 2 + 1X1 + 1X2 + 1X3 >= 1 + 1X2 active# filter(X1, X2, X3) -> active# X1 2 + 1X1 + 1X2 + 1X3 >= 1 + 1X1 active# cons(X1, X2) -> active# X1 2 + 1X1 + 0X2 >= 1 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 0 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 nats ok X -> ok nats X 0 + 0X >= 1 + 0X nats mark X -> mark nats X 0 + 0X >= 1 + 0X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 2 + 0X active zprimes() -> mark sieve nats s s 0() 2 >= 2 active nats N -> mark cons(N, nats s N) 1 + 0N >= 2 + 1N active nats X -> nats active X 1 + 0X >= 0 + 0X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 4 + 0Y + 0N >= 3 + 0Y + 0N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 0Y >= 3 + 0Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 0X >= 1 + 0X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 4 + 0Y + 1M + 1X + 0N >= 2 + 0Y + 0M + 1X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 4 + 0Y + 1M + 1X >= 3 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 4 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 1X1 + 1X2 + 1X3 >= 2 + 1X1 + 1X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: {nats# mark X -> nats# X, nats# ok X -> nats# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [sieve](x0) = x0 + 1, [nats](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [zprimes] = 1, [nats#](x0) = x0 + 1 Strict: nats# ok X -> nats# X 2 + 1X >= 1 + 1X nats# mark X -> nats# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 2 + 1X >= 2 + 1X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X active zprimes() -> mark sieve nats s s 0() 2 >= 6 active nats N -> mark cons(N, nats s N) 2 + 1N >= 2 + 1N active nats X -> nats active X 2 + 1X >= 2 + 1X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 4 + 0Y + 1N >= 3 + 0Y + 1N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 0Y >= 3 + 0Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 2 + 0Y + 1M + 0X + 0N >= 2 + 0Y + 0M + 1X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 2 + 0Y + 1M + 0X >= 3 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: {s# mark X -> s# X, s# ok X -> s# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [sieve](x0) = x0 + 1, [nats](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [zprimes] = 1, [s#](x0) = x0 + 1 Strict: s# ok X -> s# X 2 + 1X >= 1 + 1X s# mark X -> s# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 2 + 1X >= 2 + 1X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X active zprimes() -> mark sieve nats s s 0() 2 >= 6 active nats N -> mark cons(N, nats s N) 2 + 1N >= 2 + 1N active nats X -> nats active X 2 + 1X >= 2 + 1X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 4 + 0Y + 1N >= 3 + 0Y + 1N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 0Y >= 3 + 0Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 2 + 0Y + 1M + 0X + 0N >= 2 + 0Y + 0M + 1X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 2 + 0Y + 1M + 0X >= 3 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: {sieve# mark X -> sieve# X, sieve# ok X -> sieve# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [sieve](x0) = x0 + 1, [nats](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [zprimes] = 1, [sieve#](x0) = x0 + 1 Strict: sieve# ok X -> sieve# X 2 + 1X >= 1 + 1X sieve# mark X -> sieve# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 2 + 1X >= 2 + 1X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X active zprimes() -> mark sieve nats s s 0() 2 >= 6 active nats N -> mark cons(N, nats s N) 2 + 1N >= 2 + 1N active nats X -> nats active X 2 + 1X >= 2 + 1X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 4 + 0Y + 1N >= 3 + 0Y + 1N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 0Y >= 3 + 0Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 2 + 0Y + 1M + 0X + 0N >= 2 + 0Y + 0M + 1X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 2 + 0Y + 1M + 0X >= 3 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (4): Strict: { filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3), filter#(mark X1, X2, X3) -> filter#(X1, X2, X3), filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [sieve](x0) = x0 + 1, [nats](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [zprimes] = 1, [filter#](x0, x1, x2) = x0 + 1 Strict: filter#(ok X1, ok X2, ok X3) -> filter#(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 filter#(mark X1, X2, X3) -> filter#(X1, X2, X3) 2 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 filter#(X1, mark X2, X3) -> filter#(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 filter#(X1, X2, mark X3) -> filter#(X1, X2, X3) 1 + 1X1 + 0X2 + 0X3 >= 1 + 1X1 + 0X2 + 0X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 2 >= 2 proper nats X -> nats proper X 2 + 1X >= 2 + 1X proper sieve X -> sieve proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 proper 0() -> ok 0() 2 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 2 + 1X >= 2 + 1X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X active zprimes() -> mark sieve nats s s 0() 2 >= 6 active nats N -> mark cons(N, nats s N) 2 + 1N >= 2 + 1N active nats X -> nats active X 2 + 1X >= 2 + 1X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 4 + 0Y + 1N >= 3 + 0Y + 1N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 0Y >= 3 + 0Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 2 + 0Y + 1M + 0X + 0N >= 2 + 0Y + 0M + 1X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 2 + 0Y + 1M + 0X >= 3 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 SCCS (1): Scc: {filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3)} SCC (2): Strict: {filter#(X1, X2, mark X3) -> filter#(X1, X2, X3), filter#(X1, mark X2, X3) -> filter#(X1, X2, X3)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [sieve](x0) = x0 + 1, [nats](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [zprimes] = 1, [filter#](x0, x1, x2) = x0 + x1 + 1 Strict: filter#(X1, mark X2, X3) -> filter#(X1, X2, X3) 2 + 0X1 + 1X2 + 1X3 >= 1 + 0X1 + 1X2 + 1X3 filter#(X1, X2, mark X3) -> filter#(X1, X2, X3) 2 + 0X1 + 1X2 + 1X3 >= 1 + 0X1 + 1X2 + 1X3 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 2 + 1X >= 2 + 1X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X active zprimes() -> mark sieve nats s s 0() 2 >= 6 active nats N -> mark cons(N, nats s N) 2 + 1N >= 2 + 1N active nats X -> nats active X 2 + 1X >= 2 + 1X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 4 + 0Y + 1N >= 3 + 0Y + 1N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 0Y >= 3 + 0Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 2 + 0Y + 1M + 0X + 0N >= 2 + 0Y + 0M + 1X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 2 + 0Y + 1M + 0X >= 3 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = x0, [sieve](x0) = x0 + 1, [nats](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [zprimes] = 1, [cons#](x0, x1) = x0 Strict: cons#(ok X1, ok X2) -> cons#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 cons#(mark X1, X2) -> cons#(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 2 + 1X >= 2 + 1X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X active zprimes() -> mark sieve nats s s 0() 2 >= 4 active nats N -> mark cons(N, nats s N) 2 + 1N >= 2 + 1N active nats X -> nats active X 2 + 1X >= 2 + 1X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 3 + 0Y + 1N >= 2 + 0Y + 1N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 0Y >= 3 + 0Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 1 + 1X >= 1 + 1X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 2 + 0Y + 1M + 0X + 0N >= 2 + 0Y + 0M + 1X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 2 + 0Y + 1M + 0X >= 3 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 SCCS (1): Scc: {cons#(mark X1, X2) -> cons#(X1, X2)} SCC (1): Strict: {cons#(mark X1, X2) -> cons#(X1, X2)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), filter(X1, X2, mark X3) -> mark filter(X1, X2, X3), filter(X1, mark X2, X3) -> mark filter(X1, X2, X3), filter(mark X1, X2, X3) -> mark filter(X1, X2, X3), filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3), active cons(X1, X2) -> cons(active X1, X2), active filter(X1, X2, X3) -> filter(X1, X2, active X3), active filter(X1, X2, X3) -> filter(X1, active X2, X3), active filter(X1, X2, X3) -> filter(active X1, X2, X3), active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)), active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)), active s X -> s active X, active sieve X -> sieve active X, active sieve cons(0(), Y) -> mark cons(0(), sieve Y), active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)), active nats X -> nats active X, active nats N -> mark cons(N, nats s N), active zprimes() -> mark sieve nats s s 0(), s mark X -> mark s X, s ok X -> ok s X, sieve mark X -> mark sieve X, sieve ok X -> ok sieve X, nats mark X -> mark nats X, nats ok X -> ok nats X, proper cons(X1, X2) -> cons(proper X1, proper X2), proper 0() -> ok 0(), proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3), proper s X -> s proper X, proper sieve X -> sieve proper X, proper nats X -> nats proper X, proper zprimes() -> ok zprimes(), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [filter](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [sieve](x0) = x0 + 1, [nats](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 1, [zprimes] = 1, [cons#](x0, x1) = x0 + 1 Strict: cons#(mark X1, X2) -> cons#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper zprimes() -> ok zprimes() 0 >= 2 proper nats X -> nats proper X 0 + 0X >= 1 + 0X proper sieve X -> sieve proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper filter(X1, X2, X3) -> filter(proper X1, proper X2, proper X3) 0 + 0X1 + 0X2 + 0X3 >= 1 + 0X1 + 0X2 + 0X3 proper 0() -> ok 0() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 nats ok X -> ok nats X 2 + 1X >= 2 + 1X nats mark X -> mark nats X 2 + 1X >= 2 + 1X sieve ok X -> ok sieve X 2 + 1X >= 2 + 1X sieve mark X -> mark sieve X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X active zprimes() -> mark sieve nats s s 0() 2 >= 6 active nats N -> mark cons(N, nats s N) 2 + 1N >= 2 + 1N active nats X -> nats active X 2 + 1X >= 2 + 1X active sieve cons(s N, Y) -> mark cons(s N, sieve filter(Y, N, N)) 4 + 0Y + 1N >= 3 + 0Y + 1N active sieve cons(0(), Y) -> mark cons(0(), sieve Y) 4 + 0Y >= 3 + 0Y active sieve X -> sieve active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active filter(cons(X, Y), s N, M) -> mark cons(X, filter(Y, N, M)) 2 + 0Y + 1M + 0X + 0N >= 2 + 0Y + 0M + 1X + 0N active filter(cons(X, Y), 0(), M) -> mark cons(0(), filter(Y, M, M)) 2 + 0Y + 1M + 0X >= 3 + 0Y + 0M active filter(X1, X2, X3) -> filter(active X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, active X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 1 + 0X1 + 0X2 + 1X3 active filter(X1, X2, X3) -> filter(X1, X2, active X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 filter(ok X1, ok X2, ok X3) -> ok filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(mark X1, X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, mark X2, X3) -> mark filter(X1, X2, X3) 1 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 filter(X1, X2, mark X3) -> mark filter(X1, X2, X3) 2 + 0X1 + 0X2 + 1X3 >= 2 + 0X1 + 0X2 + 1X3 cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed