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) = [5] p(activate) = [5] x1 + [1] p(afterNth) = [3] x1 + [3] x2 + [6] p(cons) = [1] x2 + [0] p(fst) = [0] p(head) = [0] p(n__natsFrom) = [1] x1 + [1] p(n__s) = [1] x1 + [1] p(natsFrom) = [1] x1 + [4] p(nil) = [4] p(pair) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [4] p(sel) = [1] x2 + [1] p(snd) = [1] x1 + [2] p(splitAt) = [2] x1 + [2] x2 + [3] p(tail) = [0] p(take) = [0] p(u) = [0] p(activate#) = [5] x1 + [0] p(afterNth#) = [2] x1 + [4] x2 + [0] p(fst#) = [1] x1 + [0] p(head#) = [1] x1 + [0] p(natsFrom#) = [1] x1 + [4] p(s#) = [1] x1 + [1] p(sel#) = [3] x1 + [3] x2 + [0] p(snd#) = [1] x1 + [0] p(splitAt#) = [1] x2 + [1] p(tail#) = [6] x1 + [0] p(take#) = [5] x1 + [2] x2 + [1] p(u#) = [1] x2 + [7] x3 + [1] x4 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [4] p(c_3) = [1] x1 + [4] p(c_4) = [1] x1 + [5] p(c_5) = [4] p(c_6) = [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [4] p(c_13) = [1] x1 + [1] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [0] Following rules are strictly oriented: natsFrom#(N) = [1] N + [4] > [1] = c_7() natsFrom#(X) = [1] X + [4] > [1] = c_8() s#(X) = [1] X + [1] > [0] = c_9() activate(X) = [5] X + [1] > [1] X + [0] = X activate(n__natsFrom(X)) = [5] X + [6] > [5] X + [5] = natsFrom(activate(X)) activate(n__s(X)) = [5] X + [6] > [5] X + [5] = s(activate(X)) afterNth(N,XS) = [3] N + [3] XS + [6] > [2] N + [2] XS + [5] = snd(splitAt(N,XS)) natsFrom(N) = [1] N + [4] > [1] N + [2] = cons(N,n__natsFrom(n__s(N))) natsFrom(X) = [1] X + [4] > [1] X + [1] = n__natsFrom(X) s(X) = [1] X + [4] > [1] X + [1] = n__s(X) snd(pair(XS,YS)) = [1] XS + [1] YS + [3] > [1] YS + [0] = YS splitAt(0(),XS) = [2] XS + [13] > [1] XS + [5] = pair(nil(),XS) Following rules are (at-least) weakly oriented: activate#(X) = [5] X + [0] >= [0] = c_1() activate#(n__natsFrom(X)) = [5] X + [5] >= [5] X + [9] = c_2(natsFrom#(activate(X))) activate#(n__s(X)) = [5] X + [5] >= [5] X + [6] = c_3(s#(activate(X))) afterNth#(N,XS) = [2] N + [4] XS + [0] >= [2] N + [2] XS + [8] = c_4(snd#(splitAt(N,XS))) fst#(pair(XS,YS)) = [1] XS + [1] YS + [1] >= [4] = c_5() head#(cons(N,XS)) = [1] XS + [0] >= [0] = c_6() sel#(N,XS) = [3] N + [3] XS + [0] >= [3] N + [3] XS + [6] = c_10(head#(afterNth(N,XS))) snd#(pair(XS,YS)) = [1] XS + [1] YS + [1] >= [1] = c_11() splitAt#(0(),XS) = [1] XS + [1] >= [4] = c_12() tail#(cons(N,XS)) = [6] XS + [0] >= [5] XS + [1] = c_13(activate#(XS)) take#(N,XS) = [5] N + [2] XS + [1] >= [2] N + [2] XS + [3] = c_14(fst#(splitAt(N,XS))) u#(pair(YS,ZS),N,X,XS) = [1] N + [7] X + [1] XS + [0] >= [5] X + [0] = 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#(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() 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: natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() - 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 {1,2,3,5,6,8,9} by application of Pre({1,2,3,5,6,8,9}) = {4,7,10,11,12}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) 3: activate#(n__s(X)) -> c_3(s#(activate(X))) 4: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) 5: fst#(pair(XS,YS)) -> c_5() 6: head#(cons(N,XS)) -> c_6() 7: sel#(N,XS) -> c_10(head#(afterNth(N,XS))) 8: snd#(pair(XS,YS)) -> c_11() 9: splitAt#(0(),XS) -> c_12() 10: tail#(cons(N,XS)) -> c_13(activate#(XS)) 11: take#(N,XS) -> c_14(fst#(splitAt(N,XS))) 12: u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) 13: natsFrom#(N) -> c_7() 14: natsFrom#(X) -> c_8() 15: s#(X) -> c_9() * Step 6: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) sel#(N,XS) -> c_10(head#(afterNth(N,XS))) 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))) activate#(n__s(X)) -> c_3(s#(activate(X))) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() - 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 {1,2,3,4,5} by application of Pre({1,2,3,4,5}) = {}. Here rules are labelled as follows: 1: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) 2: sel#(N,XS) -> c_10(head#(afterNth(N,XS))) 3: tail#(cons(N,XS)) -> c_13(activate#(XS)) 4: take#(N,XS) -> c_14(fst#(splitAt(N,XS))) 5: u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) 6: activate#(X) -> c_1() 7: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) 8: activate#(n__s(X)) -> c_3(s#(activate(X))) 9: fst#(pair(XS,YS)) -> c_5() 10: head#(cons(N,XS)) -> c_6() 11: natsFrom#(N) -> c_7() 12: natsFrom#(X) -> c_8() 13: s#(X) -> c_9() 14: snd#(pair(XS,YS)) -> c_11() 15: splitAt#(0(),XS) -> c_12() * Step 7: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - 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() 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 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:W:activate#(X) -> c_1() 2:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) -->_1 natsFrom#(X) -> c_8():8 -->_1 natsFrom#(N) -> c_7():7 3:W:activate#(n__s(X)) -> c_3(s#(activate(X))) -->_1 s#(X) -> c_9():9 4:W:afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) -->_1 snd#(pair(XS,YS)) -> c_11():11 5:W:fst#(pair(XS,YS)) -> c_5() 6:W:head#(cons(N,XS)) -> c_6() 7:W:natsFrom#(N) -> c_7() 8:W:natsFrom#(X) -> c_8() 9:W:s#(X) -> c_9() 10:W:sel#(N,XS) -> c_10(head#(afterNth(N,XS))) -->_1 head#(cons(N,XS)) -> c_6():6 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))):3 -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))):2 -->_1 activate#(X) -> c_1():1 14:W:take#(N,XS) -> c_14(fst#(splitAt(N,XS))) -->_1 fst#(pair(XS,YS)) -> c_5():5 15:W:u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) -->_1 activate#(n__s(X)) -> c_3(s#(activate(X))):3 -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))):2 -->_1 activate#(X) -> c_1():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X)) 14: take#(N,XS) -> c_14(fst#(splitAt(N,XS))) 13: tail#(cons(N,XS)) -> c_13(activate#(XS)) 12: splitAt#(0(),XS) -> c_12() 10: sel#(N,XS) -> c_10(head#(afterNth(N,XS))) 6: head#(cons(N,XS)) -> c_6() 5: fst#(pair(XS,YS)) -> c_5() 4: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS))) 11: snd#(pair(XS,YS)) -> c_11() 3: activate#(n__s(X)) -> c_3(s#(activate(X))) 9: s#(X) -> c_9() 2: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))) 7: natsFrom#(N) -> c_7() 8: natsFrom#(X) -> c_8() 1: activate#(X) -> c_1() * Step 8: 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))