WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) head(cons(X,Y)) -> X sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5(s#(activate(X))) activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(n__s(X)))) from#(X) -> c_10() if#(false(),X,Y) -> c_11(activate#(Y)) if#(true(),X,Y) -> c_12(activate#(X)) primes#() -> c_13(sieve#(from(s(s(0()))))) s#(X) -> c_14() sieve#(X) -> c_15() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5(s#(activate(X))) activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(n__s(X)))) from#(X) -> c_10() if#(false(),X,Y) -> c_11(activate#(Y)) if#(true(),X,Y) -> c_12(activate#(X)) primes#() -> c_13(sieve#(from(s(s(0()))))) s#(X) -> c_14() sieve#(X) -> c_15() - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5(s#(activate(X))) activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(n__s(X)))) from#(X) -> c_10() if#(false(),X,Y) -> c_11(activate#(Y)) if#(true(),X,Y) -> c_12(activate#(X)) primes#() -> c_13(sieve#(from(s(s(0()))))) s#(X) -> c_14() sieve#(X) -> c_15() * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5(s#(activate(X))) activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(n__s(X)))) from#(X) -> c_10() if#(false(),X,Y) -> c_11(activate#(Y)) if#(true(),X,Y) -> c_12(activate#(X)) primes#() -> c_13(sieve#(from(s(s(0()))))) s#(X) -> c_14() sieve#(X) -> c_15() - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {1}, uargs(filter) = {1,2}, uargs(from) = {1}, uargs(s) = {1}, uargs(sieve) = {1}, uargs(cons#) = {1}, uargs(filter#) = {1,2}, uargs(from#) = {1}, uargs(s#) = {1}, uargs(sieve#) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(activate) = [2] x1 + [1] p(cons) = [1] x1 + [3] p(divides) = [1] x1 + [1] x2 + [0] p(false) = [0] p(filter) = [1] x1 + [1] x2 + [6] p(from) = [1] x1 + [5] p(head) = [0] p(if) = [0] p(n__cons) = [1] x1 + [2] p(n__filter) = [1] x1 + [1] x2 + [5] p(n__from) = [1] x1 + [3] p(n__s) = [1] x1 + [2] p(n__sieve) = [1] x1 + [2] p(primes) = [0] p(s) = [1] x1 + [3] p(sieve) = [1] x1 + [3] p(tail) = [0] p(true) = [0] p(activate#) = [2] x1 + [0] p(cons#) = [1] x1 + [0] p(filter#) = [1] x1 + [1] x2 + [7] p(from#) = [1] x1 + [6] p(head#) = [1] p(if#) = [2] x2 + [2] x3 + [4] p(primes#) = [0] p(s#) = [1] x1 + [4] p(sieve#) = [1] x1 + [2] p(tail#) = [2] x1 + [1] p(c_1) = [1] p(c_2) = [1] x1 + [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [2] p(c_9) = [1] x1 + [0] p(c_10) = [4] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [4] p(c_15) = [0] Following rules are strictly oriented: activate#(n__cons(X1,X2)) = [2] X1 + [4] > [2] X1 + [2] = c_2(cons#(activate(X1),X2)) activate#(n__filter(X1,X2)) = [2] X1 + [2] X2 + [10] > [2] X1 + [2] X2 + [9] = c_3(filter#(activate(X1),activate(X2))) activate#(n__sieve(X)) = [2] X + [4] > [2] X + [3] = c_6(sieve#(activate(X))) filter#(X1,X2) = [1] X1 + [1] X2 + [7] > [2] = c_8() from#(X) = [1] X + [6] > [1] X + [0] = c_9(cons#(X,n__from(n__s(X)))) from#(X) = [1] X + [6] > [4] = c_10() if#(false(),X,Y) = [2] X + [2] Y + [4] > [2] Y + [1] = c_11(activate#(Y)) if#(true(),X,Y) = [2] X + [2] Y + [4] > [2] X + [0] = c_12(activate#(X)) sieve#(X) = [1] X + [2] > [0] = c_15() activate(X) = [2] X + [1] > [1] X + [0] = X activate(n__cons(X1,X2)) = [2] X1 + [5] > [2] X1 + [4] = cons(activate(X1),X2) activate(n__filter(X1,X2)) = [2] X1 + [2] X2 + [11] > [2] X1 + [2] X2 + [8] = filter(activate(X1),activate(X2)) activate(n__from(X)) = [2] X + [7] > [2] X + [6] = from(activate(X)) activate(n__s(X)) = [2] X + [5] > [2] X + [4] = s(activate(X)) activate(n__sieve(X)) = [2] X + [5] > [2] X + [4] = sieve(activate(X)) cons(X1,X2) = [1] X1 + [3] > [1] X1 + [2] = n__cons(X1,X2) filter(X1,X2) = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [5] = n__filter(X1,X2) from(X) = [1] X + [5] > [1] X + [3] = cons(X,n__from(n__s(X))) from(X) = [1] X + [5] > [1] X + [3] = n__from(X) s(X) = [1] X + [3] > [1] X + [2] = n__s(X) sieve(X) = [1] X + [3] > [1] X + [2] = n__sieve(X) Following rules are (at-least) weakly oriented: activate#(X) = [2] X + [0] >= [1] = c_1() activate#(n__from(X)) = [2] X + [6] >= [2] X + [8] = c_4(from#(activate(X))) activate#(n__s(X)) = [2] X + [4] >= [2] X + [5] = c_5(s#(activate(X))) cons#(X1,X2) = [1] X1 + [0] >= [1] = c_7() primes#() = [0] >= [14] = c_13(sieve#(from(s(s(0()))))) s#(X) = [1] X + [4] >= [4] = c_14() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5(s#(activate(X))) cons#(X1,X2) -> c_7() primes#() -> c_13(sieve#(from(s(s(0()))))) s#(X) -> c_14() - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) filter#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(n__s(X)))) from#(X) -> c_10() if#(false(),X,Y) -> c_11(activate#(Y)) if#(true(),X,Y) -> c_12(activate#(X)) sieve#(X) -> c_15() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,6} by application of Pre({5,6}) = {3}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__from(X)) -> c_4(from#(activate(X))) 3: activate#(n__s(X)) -> c_5(s#(activate(X))) 4: cons#(X1,X2) -> c_7() 5: primes#() -> c_13(sieve#(from(s(s(0()))))) 6: s#(X) -> c_14() 7: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) 8: activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) 9: activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) 10: filter#(X1,X2) -> c_8() 11: from#(X) -> c_9(cons#(X,n__from(n__s(X)))) 12: from#(X) -> c_10() 13: if#(false(),X,Y) -> c_11(activate#(Y)) 14: if#(true(),X,Y) -> c_12(activate#(X)) 15: sieve#(X) -> c_15() * Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5(s#(activate(X))) cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) filter#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(n__s(X)))) from#(X) -> c_10() if#(false(),X,Y) -> c_11(activate#(Y)) if#(true(),X,Y) -> c_12(activate#(X)) primes#() -> c_13(sieve#(from(s(s(0()))))) s#(X) -> c_14() sieve#(X) -> c_15() - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(X) -> c_1() 2:S:activate#(n__from(X)) -> c_4(from#(activate(X))) -->_1 from#(X) -> c_9(cons#(X,n__from(n__s(X)))):9 -->_1 from#(X) -> c_10():10 3:S:activate#(n__s(X)) -> c_5(s#(activate(X))) -->_1 s#(X) -> c_14():14 4:S:cons#(X1,X2) -> c_7() 5:W:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():4 6:W:activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) -->_1 filter#(X1,X2) -> c_8():8 7:W:activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) -->_1 sieve#(X) -> c_15():15 8:W:filter#(X1,X2) -> c_8() 9:W:from#(X) -> c_9(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():4 10:W:from#(X) -> c_10() 11:W:if#(false(),X,Y) -> c_11(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_6(sieve#(activate(X))):7 -->_1 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))):6 -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)):5 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X))):3 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X))):2 -->_1 activate#(X) -> c_1():1 12:W:if#(true(),X,Y) -> c_12(activate#(X)) -->_1 activate#(n__sieve(X)) -> c_6(sieve#(activate(X))):7 -->_1 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))):6 -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)):5 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X))):3 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X))):2 -->_1 activate#(X) -> c_1():1 13:W:primes#() -> c_13(sieve#(from(s(s(0()))))) -->_1 sieve#(X) -> c_15():15 14:W:s#(X) -> c_14() 15:W:sieve#(X) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: primes#() -> c_13(sieve#(from(s(s(0()))))) 7: activate#(n__sieve(X)) -> c_6(sieve#(activate(X))) 15: sieve#(X) -> c_15() 6: activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2))) 8: filter#(X1,X2) -> c_8() 14: s#(X) -> c_14() 10: from#(X) -> c_10() * Step 7: SimplifyRHS WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5(s#(activate(X))) cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) from#(X) -> c_9(cons#(X,n__from(n__s(X)))) if#(false(),X,Y) -> c_11(activate#(Y)) if#(true(),X,Y) -> c_12(activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(X) -> c_1() 2:S:activate#(n__from(X)) -> c_4(from#(activate(X))) -->_1 from#(X) -> c_9(cons#(X,n__from(n__s(X)))):9 3:S:activate#(n__s(X)) -> c_5(s#(activate(X))) 4:S:cons#(X1,X2) -> c_7() 5:W:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():4 9:W:from#(X) -> c_9(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():4 11:W:if#(false(),X,Y) -> c_11(activate#(Y)) -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)):5 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X))):3 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X))):2 -->_1 activate#(X) -> c_1():1 12:W:if#(true(),X,Y) -> c_12(activate#(X)) -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)):5 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X))):3 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X))):2 -->_1 activate#(X) -> c_1():1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__s(X)) -> c_5() * Step 8: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5() cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) from#(X) -> c_9(cons#(X,n__from(n__s(X)))) if#(false(),X,Y) -> c_11(activate#(Y)) if#(true(),X,Y) -> c_12(activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(X) -> c_1() 2:S:activate#(n__from(X)) -> c_4(from#(activate(X))) -->_1 from#(X) -> c_9(cons#(X,n__from(n__s(X)))):6 3:S:activate#(n__s(X)) -> c_5() 4:S:cons#(X1,X2) -> c_7() 5:W:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():4 6:W:from#(X) -> c_9(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():4 7:W:if#(false(),X,Y) -> c_11(activate#(Y)) -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)):5 -->_1 activate#(n__s(X)) -> c_5():3 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X))):2 -->_1 activate#(X) -> c_1():1 8:W:if#(true(),X,Y) -> c_12(activate#(X)) -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)):5 -->_1 activate#(n__s(X)) -> c_5():3 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X))):2 -->_1 activate#(X) -> c_1():1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(7,if#(false(),X,Y) -> c_11(activate#(Y))),(8,if#(true(),X,Y) -> c_12(activate#(X)))] * Step 9: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__from(X)) -> c_4(from#(activate(X))) activate#(n__s(X)) -> c_5() cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) from#(X) -> c_9(cons#(X,n__from(n__s(X)))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(X) -> c_1() 2:S:activate#(n__from(X)) -> c_4(from#(activate(X))) -->_1 from#(X) -> c_9(cons#(X,n__from(n__s(X)))):6 3:S:activate#(n__s(X)) -> c_5() 4:S:cons#(X1,X2) -> c_7() 5:W:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():4 6:W:from#(X) -> c_9(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():4 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(1,activate#(X) -> c_1()),(3,activate#(n__s(X)) -> c_5())] * Step 10: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_4(from#(activate(X))) cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) from#(X) -> c_9(cons#(X,n__from(n__s(X)))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {}. Here rules are labelled as follows: 2: activate#(n__from(X)) -> c_4(from#(activate(X))) 4: cons#(X1,X2) -> c_7() 5: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) 6: from#(X) -> c_9(cons#(X,n__from(n__s(X)))) * Step 11: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cons#(X1,X2) -> c_7() - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) activate#(n__from(X)) -> c_4(from#(activate(X))) from#(X) -> c_9(cons#(X,n__from(n__s(X)))) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:cons#(X1,X2) -> c_7() 2:W:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2)) -->_1 cons#(X1,X2) -> c_7():1 3:W:activate#(n__from(X)) -> c_4(from#(activate(X))) -->_1 from#(X) -> c_9(cons#(X,n__from(n__s(X)))):4 4:W:from#(X) -> c_9(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():1 The dependency graph contains no loops, we remove all dependency pairs. * Step 12: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sieve(X) -> n__sieve(X) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1 ,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))