MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,filter,nats,sieve,zprimes} and constructors {0 ,cons,n__filter,n__nats,n__sieve,s} + 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#(X1,X2,X3)) activate#(n__nats(X)) -> c_3(nats#(X)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(X1,X2,X3) -> c_5() filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) nats#(N) -> c_8() nats#(X) -> c_9() sieve#(X) -> c_10() sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) activate#(n__nats(X)) -> c_3(nats#(X)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(X1,X2,X3) -> c_5() filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) nats#(N) -> c_8() nats#(X) -> c_9() sieve#(X) -> c_10() sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) zprimes() -> sieve(nats(s(s(0())))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0 ,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/2,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__sieve,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) activate#(X) -> c_1() activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) activate#(n__nats(X)) -> c_3(nats#(X)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(X1,X2,X3) -> c_5() filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) nats#(N) -> c_8() nats#(X) -> c_9() sieve#(X) -> c_10() sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) activate#(n__nats(X)) -> c_3(nats#(X)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(X1,X2,X3) -> c_5() filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) nats#(N) -> c_8() nats#(X) -> c_9() sieve#(X) -> c_10() sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0 ,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/2,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__sieve,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,8,9,10} by application of Pre({1,5,8,9,10}) = {2,3,4,6,7,11,12,13}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) 3: activate#(n__nats(X)) -> c_3(nats#(X)) 4: activate#(n__sieve(X)) -> c_4(sieve#(X)) 5: filter#(X1,X2,X3) -> c_5() 6: filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) 7: filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) 8: nats#(N) -> c_8() 9: nats#(X) -> c_9() 10: sieve#(X) -> c_10() 11: sieve#(cons(0(),Y)) -> c_11(activate#(Y)) 12: sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) 13: zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) activate#(n__nats(X)) -> c_3(nats#(X)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) - Weak DPs: activate#(X) -> c_1() filter#(X1,X2,X3) -> c_5() nats#(N) -> c_8() nats#(X) -> c_9() sieve#(X) -> c_10() - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0 ,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/2,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__sieve,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {4,5,6,7}. Here rules are labelled as follows: 1: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) 2: activate#(n__nats(X)) -> c_3(nats#(X)) 3: activate#(n__sieve(X)) -> c_4(sieve#(X)) 4: filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) 5: filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) 6: sieve#(cons(0(),Y)) -> c_11(activate#(Y)) 7: sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) 8: zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) 9: activate#(X) -> c_1() 10: filter#(X1,X2,X3) -> c_5() 11: nats#(N) -> c_8() 12: nats#(X) -> c_9() 13: sieve#(X) -> c_10() * Step 5: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) - Weak DPs: activate#(X) -> c_1() activate#(n__nats(X)) -> c_3(nats#(X)) filter#(X1,X2,X3) -> c_5() nats#(N) -> c_8() nats#(X) -> c_9() sieve#(X) -> c_10() - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0 ,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/2,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__sieve,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) -->_1 filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)):4 -->_1 filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)):3 -->_1 filter#(X1,X2,X3) -> c_5():10 2:S:activate#(n__sieve(X)) -> c_4(sieve#(X)) -->_1 sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)):6 -->_1 sieve#(cons(0(),Y)) -> c_11(activate#(Y)):5 -->_1 sieve#(X) -> c_10():13 3:S:filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) -->_1 activate#(n__nats(X)) -> c_3(nats#(X)):9 -->_1 activate#(X) -> c_1():8 -->_1 activate#(n__sieve(X)) -> c_4(sieve#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)):1 4:S:filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) -->_1 activate#(n__nats(X)) -> c_3(nats#(X)):9 -->_1 activate#(X) -> c_1():8 -->_1 activate#(n__sieve(X)) -> c_4(sieve#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)):1 5:S:sieve#(cons(0(),Y)) -> c_11(activate#(Y)) -->_1 activate#(n__nats(X)) -> c_3(nats#(X)):9 -->_1 activate#(X) -> c_1():8 -->_1 activate#(n__sieve(X)) -> c_4(sieve#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)):1 6:S:sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) -->_2 activate#(n__nats(X)) -> c_3(nats#(X)):9 -->_1 filter#(X1,X2,X3) -> c_5():10 -->_2 activate#(X) -> c_1():8 -->_1 filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)):4 -->_1 filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)):3 -->_2 activate#(n__sieve(X)) -> c_4(sieve#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)):1 7:S:zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) -->_1 sieve#(X) -> c_10():13 -->_2 nats#(X) -> c_9():12 -->_2 nats#(N) -> c_8():11 -->_1 sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)):6 -->_1 sieve#(cons(0(),Y)) -> c_11(activate#(Y)):5 8:W:activate#(X) -> c_1() 9:W:activate#(n__nats(X)) -> c_3(nats#(X)) -->_1 nats#(X) -> c_9():12 -->_1 nats#(N) -> c_8():11 10:W:filter#(X1,X2,X3) -> c_5() 11:W:nats#(N) -> c_8() 12:W:nats#(X) -> c_9() 13:W:sieve#(X) -> c_10() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: sieve#(X) -> c_10() 10: filter#(X1,X2,X3) -> c_5() 8: activate#(X) -> c_1() 9: activate#(n__nats(X)) -> c_3(nats#(X)) 11: nats#(N) -> c_8() 12: nats#(X) -> c_9() * Step 6: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0 ,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/2,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__sieve,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) -->_1 filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)):4 -->_1 filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)):3 2:S:activate#(n__sieve(X)) -> c_4(sieve#(X)) -->_1 sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)):6 -->_1 sieve#(cons(0(),Y)) -> c_11(activate#(Y)):5 3:S:filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_4(sieve#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)):1 4:S:filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_4(sieve#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)):1 5:S:sieve#(cons(0(),Y)) -> c_11(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_4(sieve#(X)):2 -->_1 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)):1 6:S:sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) -->_1 filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)):4 -->_1 filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)):3 -->_2 activate#(n__sieve(X)) -> c_4(sieve#(X)):2 -->_2 activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)):1 7:S:zprimes#() -> c_13(sieve#(nats(s(s(0())))),nats#(s(s(0())))) -->_1 sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)):6 -->_1 sieve#(cons(0(),Y)) -> c_11(activate#(Y)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: zprimes#() -> c_13(sieve#(nats(s(s(0()))))) * Step 7: NaturalMI MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) zprimes#() -> c_13(sieve#(nats(s(s(0()))))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0 ,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__sieve,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1,2}, uargs(c_13) = {1} Following symbols are considered usable: {activate#,filter#,nats#,sieve#,zprimes#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(cons) = [0] p(filter) = [0] p(n__filter) = [0] p(n__nats) = [0] p(n__sieve) = [0] p(nats) = [0] p(s) = [0] p(sieve) = [0] p(zprimes) = [0] p(activate#) = [0] p(filter#) = [0] p(nats#) = [0] p(sieve#) = [0] p(zprimes#) = [3] p(c_1) = [0] p(c_2) = [4] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [2] x1 + [0] p(c_5) = [1] p(c_6) = [4] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [4] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [2] x1 + [4] x2 + [0] p(c_13) = [1] x1 + [0] Following rules are strictly oriented: zprimes#() = [3] > [0] = c_13(sieve#(nats(s(s(0()))))) Following rules are (at-least) weakly oriented: activate#(n__filter(X1,X2,X3)) = [0] >= [0] = c_2(filter#(X1,X2,X3)) activate#(n__sieve(X)) = [0] >= [0] = c_4(sieve#(X)) filter#(cons(X,Y),0(),M) = [0] >= [0] = c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) = [0] >= [0] = c_7(activate#(Y)) sieve#(cons(0(),Y)) = [0] >= [0] = c_11(activate#(Y)) sieve#(cons(s(N),Y)) = [0] >= [0] = c_12(filter#(activate(Y),N,N),activate#(Y)) * Step 8: Failure MAYBE + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2,X3)) -> c_2(filter#(X1,X2,X3)) activate#(n__sieve(X)) -> c_4(sieve#(X)) filter#(cons(X,Y),0(),M) -> c_6(activate#(Y)) filter#(cons(X,Y),s(N),M) -> c_7(activate#(Y)) sieve#(cons(0(),Y)) -> c_11(activate#(Y)) sieve#(cons(s(N),Y)) -> c_12(filter#(activate(Y),N,N),activate#(Y)) - Weak DPs: zprimes#() -> c_13(sieve#(nats(s(s(0()))))) - Weak TRS: activate(X) -> X activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3) activate(n__nats(X)) -> nats(X) activate(n__sieve(X)) -> sieve(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(s(N))) nats(X) -> n__nats(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(filter(activate(Y),N,N))) - Signature: {activate/1,filter/3,nats/1,sieve/1,zprimes/0,activate#/1,filter#/3,nats#/1,sieve#/1,zprimes#/0} / {0/0 ,cons/2,n__filter/3,n__nats/1,n__sieve/1,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,filter#,nats#,sieve# ,zprimes#} and constructors {0,cons,n__filter,n__nats,n__sieve,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE