MAYBE * Step 1: InnermostRuleRemoval MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,filter,nats,s,sieve,zprimes} and constructors {0 ,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M)) sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N))) All above mentioned rules can be savely removed. * Step 2: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,filter,nats,s,sieve,zprimes} and constructors {0 ,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(X1,X2,X3) -> c_6() filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) nats#(N) -> c_8() nats#(X) -> c_9() s#(X) -> c_10() sieve#(X) -> c_11() sieve#(cons(0(),Y)) -> c_12(activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) Weak DPs and mark the set of starting terms. * Step 3: UsableRules MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(X1,X2,X3) -> c_6() filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) nats#(N) -> c_8() nats#(X) -> c_9() s#(X) -> c_10() sieve#(X) -> c_11() sieve#(cons(0(),Y)) -> c_12(activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/2,c_4/2,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) activate#(X) -> c_1() activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(X1,X2,X3) -> c_6() filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) nats#(N) -> c_8() nats#(X) -> c_9() s#(X) -> c_10() sieve#(X) -> c_11() sieve#(cons(0(),Y)) -> c_12(activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(X1,X2,X3) -> c_6() filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) nats#(N) -> c_8() nats#(X) -> c_9() s#(X) -> c_10() sieve#(X) -> c_11() sieve#(cons(0(),Y)) -> c_12(activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/2,c_4/2,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,6,8,9,10,11} by application of Pre({1,6,8,9,10,11}) = {2,3,4,5,7,12,13}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) 3: activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) 4: activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) 5: activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) 6: filter#(X1,X2,X3) -> c_6() 7: filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) 8: nats#(N) -> c_8() 9: nats#(X) -> c_9() 10: s#(X) -> c_10() 11: sieve#(X) -> c_11() 12: sieve#(cons(0(),Y)) -> c_12(activate#(Y)) 13: zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) * Step 5: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_12(activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) - Weak DPs: activate#(X) -> c_1() filter#(X1,X2,X3) -> c_6() nats#(N) -> c_8() nats#(X) -> c_9() s#(X) -> c_10() sieve#(X) -> c_11() - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/2,c_4/2,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) -->_1 filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)):5 -->_4 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_3 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_4 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_3 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_4 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_3 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_2 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_1 filter#(X1,X2,X3) -> c_6():9 -->_4 activate#(X) -> c_1():8 -->_3 activate#(X) -> c_1():8 -->_2 activate#(X) -> c_1():8 -->_4 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 -->_3 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 2:S:activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_1 nats#(X) -> c_9():11 -->_1 nats#(N) -> c_8():10 -->_2 activate#(X) -> c_1():8 -->_2 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 3:S:activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 s#(X) -> c_10():12 -->_2 activate#(X) -> c_1():8 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 4:S:activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) -->_1 sieve#(cons(0(),Y)) -> c_12(activate#(Y)):6 -->_1 sieve#(X) -> c_11():13 -->_2 activate#(X) -> c_1():8 -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 5:S:filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) -->_1 activate#(X) -> c_1():8 -->_1 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_1 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 6:S:sieve#(cons(0(),Y)) -> c_12(activate#(Y)) -->_1 activate#(X) -> c_1():8 -->_1 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_1 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 7:S:zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) -->_1 sieve#(X) -> c_11():13 -->_4 s#(X) -> c_10():12 -->_3 s#(X) -> c_10():12 -->_2 nats#(X) -> c_9():11 -->_2 nats#(N) -> c_8():10 -->_1 sieve#(cons(0(),Y)) -> c_12(activate#(Y)):6 8:W:activate#(X) -> c_1() 9:W:filter#(X1,X2,X3) -> c_6() 10:W:nats#(N) -> c_8() 11:W:nats#(X) -> c_9() 12:W:s#(X) -> c_10() 13:W:sieve#(X) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: filter#(X1,X2,X3) -> c_6() 13: sieve#(X) -> c_11() 10: nats#(N) -> c_8() 11: nats#(X) -> c_9() 12: s#(X) -> c_10() 8: activate#(X) -> c_1() * Step 6: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_12(activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/2,c_4/2,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) -->_1 filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)):5 -->_4 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_3 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_4 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_3 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_4 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_3 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_2 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_4 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 -->_3 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 2:S:activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)) -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 3:S:activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 4:S:activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) -->_1 sieve#(cons(0(),Y)) -> c_12(activate#(Y)):6 -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 5:S:filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_1 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 6:S:sieve#(cons(0(),Y)) -> c_12(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_1 activate#(n__nats(X)) -> c_3(nats#(activate(X)),activate#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 7:S:zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0()))),s#(s(0())),s#(0())) -->_1 sieve#(cons(0(),Y)) -> c_12(activate#(Y)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__nats(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) zprimes#() -> c_13(sieve#(nats(s(s(0()))))) * Step 7: Decompose MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_12(activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0()))))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/1,c_4/1,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_12(activate#(Y)) - Weak DPs: zprimes#() -> c_13(sieve#(nats(s(s(0()))))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/1,c_4/1,c_5/2,c_6/0 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} Problem (S) - Strict DPs: zprimes#() -> c_13(sieve#(nats(s(s(0()))))) - Weak DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_12(activate#(Y)) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/1,c_4/1,c_5/2,c_6/0 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} ** Step 7.a:1: Failure MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_12(activate#(Y)) - Weak DPs: zprimes#() -> c_13(sieve#(nats(s(s(0()))))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/1,c_4/1,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 7.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: zprimes#() -> c_13(sieve#(nats(s(s(0()))))) - Weak DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_12(activate#(Y)) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/1,c_4/1,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: zprimes#() -> c_13(sieve#(nats(s(s(0()))))) 2: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) 3: activate#(n__nats(X)) -> c_3(activate#(X)) 4: activate#(n__s(X)) -> c_4(activate#(X)) 5: activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) 6: filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) 7: sieve#(cons(0(),Y)) -> c_12(activate#(Y)) ** Step 7.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) activate#(n__nats(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_12(activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0()))))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/1,c_4/1,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) -->_1 filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)):5 -->_4 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_3 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_4 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_3 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_2 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_4 activate#(n__nats(X)) -> c_3(activate#(X)):2 -->_3 activate#(n__nats(X)) -> c_3(activate#(X)):2 -->_2 activate#(n__nats(X)) -> c_3(activate#(X)):2 -->_4 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 -->_3 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 2:W:activate#(n__nats(X)) -> c_3(activate#(X)) -->_1 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__nats(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 3:W:activate#(n__s(X)) -> c_4(activate#(X)) -->_1 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__nats(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 4:W:activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) -->_1 sieve#(cons(0(),Y)) -> c_12(activate#(Y)):6 -->_2 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_2 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_2 activate#(n__nats(X)) -> c_3(activate#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 5:W:filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__nats(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 6:W:sieve#(cons(0(),Y)) -> c_12(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)):4 -->_1 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__nats(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)):1 7:W:zprimes#() -> c_13(sieve#(nats(s(s(0()))))) -->_1 sieve#(cons(0(),Y)) -> c_12(activate#(Y)):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: zprimes#() -> c_13(sieve#(nats(s(s(0()))))) 1: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(activate(X1),activate(X2),activate(X3)) ,activate#(X1) ,activate#(X2) ,activate#(X3)) 6: sieve#(cons(0(),Y)) -> c_12(activate#(Y)) 4: activate#(n__sieve(X)) -> c_5(sieve#(activate(X)),activate#(X)) 5: filter#(cons(X,Y),0(),M) -> c_7(activate#(Y)) 3: activate#(n__s(X)) -> c_4(activate#(X)) 2: activate#(n__nats(X)) -> c_3(activate#(X)) ** Step 7.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3)) activate(n__nats(X)) -> nats(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) filter(X1,X2,X3) -> n__filter(X1,X2,X3) filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M)) nats(N) -> cons(N,n__nats(n__s(N))) nats(X) -> n__nats(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y))) - Signature: {activate/1,filter/3,nats/1,s/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,s#/1,sieve#/1 ,zprimes#/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__s/1,n__sieve/1,c_1/0,c_2/4,c_3/1,c_4/1,c_5/2,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,s#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__s,n__sieve} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE