WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt ,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) 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__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt ,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs activate#(X) -> c_1() activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) activate#(n__s(X)) -> c_3(s#(activate(X))) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS))) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() tail#(cons(N,XS)) -> c_13(activate#(XS)) take#(N,XS) -> c_14(fst#(splitAt(N,XS))) u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) 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__natsFrom(X)) -> c_2(natsFrom#(activate(X))) activate#(n__s(X)) -> c_3(s#(activate(X))) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS))) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() tail#(cons(N,XS)) -> c_13(activate#(XS)) take#(N,XS) -> c_14(fst#(splitAt(N,XS))) u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) - Strict TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) activate#(X) -> c_1() activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) activate#(n__s(X)) -> c_3(s#(activate(X))) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS))) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() tail#(cons(N,XS)) -> c_13(activate#(XS)) take#(N,XS) -> c_14(fst#(splitAt(N,XS))) u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) activate#(n__s(X)) -> c_3(s#(activate(X))) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS))) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() tail#(cons(N,XS)) -> c_13(activate#(XS)) take#(N,XS) -> c_14(fst#(splitAt(N,XS))) u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) - Strict TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + 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(natsFrom) = {1}, uargs(s) = {1}, uargs(snd) = {1}, uargs(fst#) = {1}, uargs(head#) = {1}, uargs(natsFrom#) = {1}, uargs(s#) = {1}, uargs(snd#) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_10) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_15) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [2] x1 + [4] p(afterNth) = [4] x2 + [5] p(cons) = [1] x2 + [0] p(fst) = [2] x1 + [0] p(head) = [1] x1 + [1] p(n__natsFrom) = [1] x1 + [4] p(n__s) = [1] x1 + [2] p(natsFrom) = [1] x1 + [7] p(nil) = [0] p(pair) = [1] x2 + [0] p(s) = [1] x1 + [3] p(sel) = [0] p(snd) = [1] x1 + [3] p(splitAt) = [2] x2 + [1] p(tail) = [0] p(take) = [0] p(u) = [4] x2 + [2] x3 + [0] p(activate#) = [2] x1 + [1] p(afterNth#) = [3] x2 + [0] p(fst#) = [1] x1 + [3] p(head#) = [1] x1 + [4] p(natsFrom#) = [1] x1 + [0] p(s#) = [1] x1 + [1] p(sel#) = [1] x1 + [5] x2 + [2] p(snd#) = [1] x1 + [4] p(splitAt#) = [0] p(tail#) = [2] x1 + [0] p(take#) = [2] x1 + [3] x2 + [0] p(u#) = [2] x3 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [0] Following rules are strictly oriented: activate#(X) = [2] X + [1] > [0] = c_1() activate#(n__natsFrom(X)) = [2] X + [9] > [2] X + [4] = c_2(natsFrom#(activate(X))) fst#(pair(XS,YS)) = [1] YS + [3] > [0] = c_5() head#(cons(N,XS)) = [1] XS + [4] > [0] = c_6() s#(X) = [1] X + [1] > [0] = c_9() snd#(pair(XS,YS)) = [1] YS + [4] > [0] = c_11() activate(X) = [2] X + [4] > [1] X + [0] = X activate(n__natsFrom(X)) = [2] X + [12] > [2] X + [11] = natsFrom(activate(X)) activate(n__s(X)) = [2] X + [8] > [2] X + [7] = s(activate(X)) afterNth(N,XS) = [4] XS + [5] > [2] XS + [4] = snd(splitAt(N,XS)) natsFrom(N) = [1] N + [7] > [1] N + [6] = cons(N,n__natsFrom(n__s(N))) natsFrom(X) = [1] X + [7] > [1] X + [4] = n__natsFrom(X) s(X) = [1] X + [3] > [1] X + [2] = n__s(X) snd(pair(XS,YS)) = [1] YS + [3] > [1] YS + [0] = YS splitAt(0(),XS) = [2] XS + [1] > [1] XS + [0] = pair(nil(),XS) Following rules are (at-least) weakly oriented: activate#(n__s(X)) = [2] X + [5] >= [2] X + [5] = c_3(s#(activate(X))) afterNth#(N,XS) = [3] XS + [0] >= [2] XS + [5] = c_4(snd#(splitAt(N,XS))) natsFrom#(N) = [1] N + [0] >= [0] = c_7() natsFrom#(X) = [1] X + [0] >= [0] = c_8() sel#(N,XS) = [1] N + [5] XS + [2] >= [4] XS + [9] = c_10(head#(afterNth(N,XS))) splitAt#(0(),XS) = [0] >= [0] = c_12() tail#(cons(N,XS)) = [2] XS + [0] >= [2] XS + [1] = c_13(activate#(XS)) take#(N,XS) = [2] N + [3] XS + [0] >= [2] XS + [4] = c_14(fst#(splitAt(N,XS))) u#(pair(YS,ZS),N,X,XS) = [2] X + [0] >= [2] X + [1] = c_15(activate#(X)) 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#(n__s(X)) -> c_3(s#(activate(X))) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() sel#(N,XS) -> c_10(head#(afterNth(N,XS))) splitAt#(0(),XS) -> c_12() tail#(cons(N,XS)) -> c_13(activate#(XS)) take#(N,XS) -> c_14(fst#(splitAt(N,XS))) u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) - Weak DPs: activate#(X) -> c_1() activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() s#(X) -> c_9() snd#(pair(XS,YS)) -> c_11() - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,5,6,8} by application of Pre({1,2,5,6,8}) = {7,9}. Here rules are labelled as follows: 1: activate#(n__s(X)) -> c_3(s#(activate(X))) 2: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) 3: natsFrom#(N) -> c_7() 4: natsFrom#(X) -> c_8() 5: sel#(N,XS) -> c_10(head#(afterNth(N,XS))) 6: splitAt#(0(),XS) -> c_12() 7: tail#(cons(N,XS)) -> c_13(activate#(XS)) 8: take#(N,XS) -> c_14(fst#(splitAt(N,XS))) 9: u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) 10: activate#(X) -> c_1() 11: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) 12: fst#(pair(XS,YS)) -> c_5() 13: head#(cons(N,XS)) -> c_6() 14: s#(X) -> c_9() 15: snd#(pair(XS,YS)) -> c_11() * Step 6: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() tail#(cons(N,XS)) -> c_13(activate#(XS)) u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) - Weak DPs: activate#(X) -> c_1() activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) activate#(n__s(X)) -> c_3(s#(activate(X))) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS))) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() take#(N,XS) -> c_14(fst#(splitAt(N,XS))) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,4} by application of Pre({3,4}) = {}. Here rules are labelled as follows: 1: natsFrom#(N) -> c_7() 2: natsFrom#(X) -> c_8() 3: tail#(cons(N,XS)) -> c_13(activate#(XS)) 4: u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) 5: activate#(X) -> c_1() 6: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) 7: activate#(n__s(X)) -> c_3(s#(activate(X))) 8: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) 9: fst#(pair(XS,YS)) -> c_5() 10: head#(cons(N,XS)) -> c_6() 11: s#(X) -> c_9() 12: sel#(N,XS) -> c_10(head#(afterNth(N,XS))) 13: snd#(pair(XS,YS)) -> c_11() 14: splitAt#(0(),XS) -> c_12() 15: take#(N,XS) -> c_14(fst#(splitAt(N,XS))) * Step 7: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() - Weak DPs: activate#(X) -> c_1() activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) activate#(n__s(X)) -> c_3(s#(activate(X))) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS))) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() tail#(cons(N,XS)) -> c_13(activate#(XS)) take#(N,XS) -> c_14(fst#(splitAt(N,XS))) u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:natsFrom#(N) -> c_7() 2:S:natsFrom#(X) -> c_8() 3:W:activate#(X) -> c_1() 4:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) -->_1 natsFrom#(X) -> c_8():2 -->_1 natsFrom#(N) -> c_7():1 5:W:activate#(n__s(X)) -> c_3(s#(activate(X))) -->_1 s#(X) -> c_9():9 6:W:afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) -->_1 snd#(pair(XS,YS)) -> c_11():11 7:W:fst#(pair(XS,YS)) -> c_5() 8:W:head#(cons(N,XS)) -> c_6() 9:W:s#(X) -> c_9() 10:W:sel#(N,XS) -> c_10(head#(afterNth(N,XS))) -->_1 head#(cons(N,XS)) -> c_6():8 11:W:snd#(pair(XS,YS)) -> c_11() 12:W:splitAt#(0(),XS) -> c_12() 13:W:tail#(cons(N,XS)) -> c_13(activate#(XS)) -->_1 activate#(n__s(X)) -> c_3(s#(activate(X))):5 -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))):4 -->_1 activate#(X) -> c_1():3 14:W:take#(N,XS) -> c_14(fst#(splitAt(N,XS))) -->_1 fst#(pair(XS,YS)) -> c_5():7 15:W:u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) -->_1 activate#(n__s(X)) -> c_3(s#(activate(X))):5 -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))):4 -->_1 activate#(X) -> c_1():3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: take#(N,XS) -> c_14(fst#(splitAt(N,XS))) 12: splitAt#(0(),XS) -> c_12() 10: sel#(N,XS) -> c_10(head#(afterNth(N,XS))) 8: head#(cons(N,XS)) -> c_6() 7: fst#(pair(XS,YS)) -> c_5() 6: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) 11: snd#(pair(XS,YS)) -> c_11() 5: activate#(n__s(X)) -> c_3(s#(activate(X))) 9: s#(X) -> c_9() 3: activate#(X) -> c_1() * Step 8: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() - Weak DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) tail#(cons(N,XS)) -> c_13(activate#(XS)) u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:natsFrom#(N) -> c_7() 2:S:natsFrom#(X) -> c_8() 4:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) -->_1 natsFrom#(X) -> c_8():2 -->_1 natsFrom#(N) -> c_7():1 13:W:tail#(cons(N,XS)) -> c_13(activate#(XS)) -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))):4 15:W:u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))):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). [(13,tail#(cons(N,XS)) -> c_13(activate#(XS))),(15,u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)))] * Step 9: Decompose WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() - Weak DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + 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: natsFrom#(N) -> c_7() - Weak DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) natsFrom#(X) -> c_8() - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} Problem (S) - Strict DPs: natsFrom#(X) -> c_8() - Weak DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) natsFrom#(N) -> c_7() - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} ** Step 9.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: natsFrom#(N) -> c_7() - Weak DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) natsFrom#(X) -> c_8() - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:natsFrom#(N) -> c_7() 2:W:natsFrom#(X) -> c_8() 4:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) -->_1 natsFrom#(N) -> c_7():1 -->_1 natsFrom#(X) -> c_8():2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: natsFrom#(X) -> c_8() ** Step 9.a:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: natsFrom#(N) -> c_7() - Weak DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:natsFrom#(N) -> c_7() 4:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) -->_1 natsFrom#(N) -> c_7():1 The dependency graph contains no loops, we remove all dependency pairs. ** Step 9.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 9.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: natsFrom#(X) -> c_8() - Weak DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) natsFrom#(N) -> c_7() - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:natsFrom#(X) -> c_8() 2:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) -->_1 natsFrom#(N) -> c_7():3 -->_1 natsFrom#(X) -> c_8():1 3:W:natsFrom#(N) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: natsFrom#(N) -> c_7() ** Step 9.b:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: natsFrom#(X) -> c_8() - Weak DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:natsFrom#(X) -> c_8() 2:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) -->_1 natsFrom#(X) -> c_8():1 The dependency graph contains no loops, we remove all dependency pairs. ** Step 9.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))